Hybrid Tail Risk and Expected Stock Returns: When Does...

Hybrid Tail Risk and Expected Stock Returns:

When Does the Tail Wag the Dog?

Turan G. Bali

Georgetown University

Nusret Cakici

Fordham University

Robert F. Whitelaw

New York University and NBER

We introduce a new hybrid measure of stock return tail covariance risk, motivated by the

underdiversified portfolio holdings of individual investors, and investigate its cross-sec-

tional predictive power. Our key innovation is that this covariance is measured across the

left tail states of the individual stock return distribution, and not across those of themarket

return as in standard systematic risk measures. We document a positive and significant

relation between hybrid tail covariance risk (H-TCR) and expected stock returns, with an

annualized premium of 9%, in contrast to the insignificant or negative results for purely

stock-specific or systematic tail risk measures. (JEL G10, G11, C13)

In spite of the dominance of the CAPM paradigm, there has been a long-standing interest in the literature regarding the question of whether downsideor tail risk plays a special role in determining expected returns. Such a rolecould come about, for example, because of preferences that treat losses andgains asymmetrically,1 return distributions that are asymmetric, or somecombination of the two. While systematic downside or tail risk is a naturalstarting point, there is increasing evidence that nonmarket risk may play animportant role in determining the cross-section of expected returns.2 Thus, we

Wewould like to thank the editor, Jeff Pontiff, and twoanonymous referees for their extremely helpful commentsand suggestions.We also benefited from discussions with seminar participants at GeorgetownUniversity, HongKong University of Science and Technology, and the University of Utah. Send correspondence to Robert F.Whitelaw, Stern School of Business, NewYorkUniversity, 44W. 4th Street, Suite 9-190, NewYork, NY 10012,USA; telephone: (212) 998-0338. E-mail: [email protected].

1 See, for example,Kahneman,Knetsch, andThaler (1990) for some of the extensive experimental evidence on lossaversion.

2 For recent examples, see Ang et al. (2006, 2009) and Bali, Cakici and Whitelaw (2011).

� The Author 2014. Published by Oxford University Press on behalf of The Society for Financial Studies.All rights reserved. For Permissions, please email: [email protected]:10.1093/rapstu/rau006 Advance Access publication September 15, 2014

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consider a setting in which investors hold concentrated stock holdings in

addition to a fraction of their wealth in a well-diversified portfolio, for ex-

ample, a mutual fund within a retirement account, consistent with existing

empirical evidence on the holdings of individual investors.3 In this setting the

contribution of an individual stock to the tail risk of the portfolio can be

decomposed into three components: a systematic component, a stock-specific

component, and a hybrid component that depends on the cotail risk of the

stock and the market portfolio. Based on this decomposition, we conduct a

thorough re-examination of the role of downside risk in determining the

cross-section of expected returns. Specifically, controlling for the usual deter-

minants of expected returns, we investigate the predictive power of various

downside risk measures that vary across two dimensions: (1) the fraction of

the lower half of the return distribution that they measure and on which they

are calculated, that is, the extent to which they are tail risk measures, and (2)

the extent to which they capture systematic versus idiosyncratic or stock-

specific risk.Our risk measures build on the notion of semivariance or the lower partial

moment (LPM) of Markowitz (1959). The LPM of an asset or portfolio is

defined as

LPM ¼

Zh

�1

ðR� hÞ2fpðRÞ dR; ð1Þ

where h is the target level of returns and fpðRÞ represents the probability

density function of returns for portfolio p. That is, the semivariance is the

expected value of the squared negative deviations from themean, whereas the

more general LPM uses a chosen point of reference (h). The main heuristic

motivation for the use of the LPM in place of variance as a measure of risk is

that the LPM measures losses (relative to some reference point), whereas

variance depends on gains and losses. Of course, for symmetric distributions

and a reference point equal to the mean of the distribution, this distinction is

meaningless.In our simplified setting in which investors hold an underdiversified port-

folio consisting of positions in an individual stock and a diversified fund, three

factors contribute to the tail risk of the portfolio. First, systematic tail risk

matters in that the tail risk of the market portfolio contributes to the tail risk

of the overall portfolio. We take our systematic risk measure from the mean-

lower partial moment CAPM of Bawa and Lindenberg (1977):

�LPM ¼Eð½Ri � h�½Rm � h�jRm < hÞ

Eð½Rm � h�2jRm < hÞ; ð2Þ

3 See Polkovnichenko (2005) and Van Nieuwerburgh and Veldkamp (2010).

Hybrid Tail Risk and Expected Stock Returns

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where EðRiÞ is the expected return on asset i, EðRmÞ is the expected return onthe market portfolio, rf is the risk-free interest rate, and �LPM is a measure of

downside systematic risk for a target level of returns h. That is, the relevantbeta in the model is the colower partial moment of the asset return with themarket return divided by the LPM of the market return, where the momentsare conditional on the market return being below a specified threshold.Earlier studies on this model use alternative measures of downside market

risk based on different return thresholds, such as the mean excess marketreturn, the risk-free rate, or zero.4More recently, Ang, Chen, andXing (2006)re-examine these downside betas. Motivated by the possibility that it is moreextreme negative realizations about which investors care or that it is asym-metries in the tail of return distributions that are important, we examine

alternative measures of downside beta based on the observations in thelower tail of the market return distribution. There is recent evidence thatsystematic crash risk is priced in the cross-section of expected returns(Kelly and Jiang 2013; Ruenzi and Weigert 2013), but these studies considerinfrequent events of extreme magnitude, in the spirit of the rare disastermodels of Rietz (1988) and Barro (2006), using copula-based methods andempirical techniques from extreme value theory. In contrast, we consider the

more frequent but less extreme tail events that occur on a regular basis, usingmore traditional risk measures.5

Second, it is clear that the tail risk of the individual stock will also matterfor the tail risk of the underdiversified portfolio. For stock-specific tail risk,we use the LPM of individual stock returns.Finally, we propose a new, hybrid measure of tail risk. Given that individ-

ual stocks generally have substantially higher volatilities than do the marketportfolio and assuming a sufficient weight in the stock in the portfolio, the tailevents for such an underdiversified portfolio will coincide more with the tailevents of the individual stocks than with the tail events of the diversifiedholdings. Thus, we construct a measure called hybrid tail covariance risk

(H-TCR), which is defined as6

H-TCRi ¼ Eð½Ri � hi�½Rm � hm�jRi < hiÞ; ð3Þ

where h denotes the return threshold, for example, the 10th percentile of thereturn distribution of the stock or market. Hybrid tail covariance risk(H-TCR) is the colower partial moment between extreme daily returns on

4 See, for example, Jahankhani (1976), Price, Price, and Nantell (1982), and Harlow and Rao (1989).

5 There are several reasons why empirical studies consider measures of downside risk in the cross-sectional pricingof individual stocks. First, introduced by Roy (1952), Telser (1955), and Baumol (1963), there is a long line oftheoretical literature about safety-first investors whominimize the chance of disaster. Second,Markowitz (1959)proposed semivariance as an alternative measure of portfolio risk. Finally, for expected utility maximizinginvestors, Bawa (1975) provides a theoretical rationale for using semivariance or the lower partial moment asthe measure of portfolio risk.

6 We motivate H-TCR more formally in the context of a stylized model in the next section.

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stock i and the corresponding daily returns on the market portfolio, condi-tional on the stock return being below the specified threshold. H-TCR isanalogous to the numerator of the beta defined in Equation (2), except thatthe moment is conditional on the return on the individual stock rather thanon the return on the market.

In our empirical analysis, we compute the above measures of tail risk(systematic, hybrid, and stock-specific) for individual stocks using sixmonths and one year of daily data. We then ask which, if any, of thesemeasures have predictive power for returns over the subsequent monthusing NYSE/AMEX/NASDAQ stocks for the July 1963–December 2012sample period. In addition to the standard controls in cross-sectional tests,we are also careful to control for volatility (Ang et al. 2006, 2009) and extremereturns (Bali, Cakici, and Whitelaw 2011) because these stock-specific distri-butional characteristics are likely to be correlatedwith both the LPMof stockreturns and our hybrid measure of tail risk.

The results are striking. First, systematic risk has little or no explanatorypower for future returns, whethermeasured relative to the center or the tail ofthe distribution. This evidence appears to contradict the crash risk results inKelly and Jiang (2013) and Ruenzi andWeigert (2013) and the downside riskresults in Ang, Chen, and Xing (2006). In the former case, one natural inter-pretation is that tail risk is only priced very deep in the tails. Therefore, be-cause of the scarcity of data, one needs sophisticated empirical methodologiesto uncover this phenomenon. In the latter case, we show that the existingresults in the literature are not robust to changes in the sample selectionmethodology, thereby casting doubt on their general validity.

Second, stock-specific risk is, if anything, priced negatively, that is, in theopposite direction of that implied by theory. However, these results should beinterpretedwith caution because of the difficulty of distinguishing any tail riskeffect from the pricing of other distributional characteristics.

Third, andmost important, inmarked contrast to these results,H-TCRhassignificant and robust positive predictive power for future returns. Univariateportfolio level analyses indicate that a trading strategy that goes long stocks inthe highest H-TCRdecile and shorts stocks in the lowest H-TCRdecile yieldsaverage raw and risk-adjusted returns of up to 9% per annum. Firm-level,cross-sectional regressions that control for well-known pricing effects, includ-ing size, book-to-market (Fama and French 1992, 1993), momentum(Jegadeesh andTitman 1993), short-term reversals (Jegadeesh 1990), liquidity(Amihud 2002), coskewness (Harvey and Siddique 1999, 2000), downsidebeta (Ang, Chen, and Xing 2006), volatility (Ang et al. 2006, 2009), netshare issuance (Pontiff and Woodgate 2008), and preference for lottery-likeassets (Bali, Cakici, and Whitelaw 2011) generate similar results. Moreover,there is strong evidence that the pricing of H-TCR is a tail risk, rather than amore general downside risk phenomenon, as the effect attenuates significantlyas the fraction of observations used to calculate the measure increases.


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One might argue that our failure to detect significant pricing of systematicand stock-specific tail risk calls into question the interpretation of the signifi-cance of our hybrid measure. To the contrary, we view the success of thehybridmeasure as convincing evidence that tail risk doesmatter.H-TCR is anideal variable because it is not highly correlated with other cross-sectionalpredictors, yet it is an important determinant of the tail risk of concentratedportfolios. Stock-specific tail risk does not meet the first criterion because ofits high correlation with other stock-specific variables, such as volatility, andsystematic tail risk does not meet the second criterion because of its relativelyweak link to cross-sectional variation in the tail risk of underdiversified port-folios. Of course, given these issues, it is all the more important to demon-strate the robustness of H-TCR.As robustness checks, we test whether the positive relation between tail

covariance risk and the cross-section of expected returns holds in bivariatedependent sorts, using size- and book-to-market-matched benchmark port-folios similar to Daniel and Titman (1997), and once we screen for extremestocks across numerous dimensions. Throughout our empirical analysis, theevidence is consistent with significant pricing effects generated by individualinvestors who care about how the tail risk of their concentrated positionsinteracts with their diversified holdings.

1. Motivating Theory and Empirical Evidence

To motivate our three measures of tail risk, we develop a relatively stylized,1-period, discrete state space model in which systematic, stock-specific, andhybrid tail risk arise as appropriate measures of risk for an individual stock.Specifically, these three variables capture the extent to which an individualstock contributes to the tail risk of an underdiversified portfolio, where theform of the portfolio is motivated by existing empirical evidence on the stockholdings of individual investors. We also present empirical evidence on thecross-sectional relation between these measures and the tail risk of concen-trated portfolios. Of course, for these tail risk variables to be priced in equi-librium, it is also necessary that underdiversified investors care about tail riskand that these investors influence pricing. Specifying a general equilibriummodel in which this is the case is beyond the scope of this paper. Instead, weappeal to the long line of literature on both tail risk and limits to arbitrage tomotivate the idea that ultimately it is an empirical question.Although diversification is critical in eliminating idiosyncratic risk, a

closer examination of the portfolios of individual investors suggeststhat these investors are, in general, not well diversified.7 For example,

7 It is important to note that this underdiversification relative to the implications of classical models of portfoliochoice could be completely consistentwith rationality inmore complexmodels. For example, Roche, Tompaidis,andYang (2013), VanNieuwerburgh andVeldkamp (2010), and references therein propose a number of rationalmodels that generate concentrated holdings.


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Polkovnichenko (2005) examines a survey of fourteenmillion households andshows that the median number of stocks in household portfolios is two in

1989, 1992, 1995, and 1998. The median increases to three stocks in 2001.Based on 40,000 stock accounts at a brokerage firm, Goetzmann andKumar(2008) find that the median number of stocks in a portfolio of individualinvestors is three for the 1991–1996 period. These results are similar to thefindings of earlier studies. For example, Blume and Friend (1975) and Blume,Crockett, and Friend (1974) provide evidence that the average number of

stocks in household portfolios is about 3.41 in 1967.Odean (1999) andBarberand Odean (2001) also report the median number of stocks in individualinvestors’ portfolios as two to three. In recent work, Dorn and Huberman(2005, 2010) use trading records between 1995 and 2000 of over 20,000 cus-tomers of a German discount brokerage and find that the typical portfolio

consists of little more than three stocks.However, these individual stock holdings often do not constitute the full

financial asset portfolios of these investors. Polkovnichenko (2005) reportsthe fraction of individual equities relative to total financial assets as 33% in1989, 39% in 1992, 49% in 1995, 53% in 1998, and 57% in 2001. That is,investors have a significant fraction of their wealth in concentrated holdings,but they also hold wealth in other investments that may take the form of, for

example, diversified mutual funds in retirement accounts.Based on this evidence, consider an investor that holds a portfolio con-

sisting of positions in two assets—equity in an individual firm and themarket portfolio. Assume that over the next period the returns on thefirm (Ri) and market (Rm) take on J and K discrete values, respectively,indexed by j¼ 1, . . . , J and k¼ 1, . . . , K and in order of increasing returns.That is, the return in state j on firm i is greater than the return in state j-1(Ri,j>Ri,j-1) and similarly for the market (Rm,k>Rm,k-1). There are J x K

possible states of the world, each occurring with probability pjk, where theprobability of a given state can be zero. Denote the investor’s nonnegativeportfolio weights in the two assets as wi and wm, where wi þ wm ¼ 1. Theportfolio return in each state is

RP;jk ¼ wiRi;j þ ð1� wiÞRm;k: ð4Þ

Now assume further that the relevant measure of risk is the LPM of the

portfolio, as defined in Equation (1), for a specified threshold h. This calcu-

lation requires an ordering of the J x K states in terms of the associated

portfolio return in order to compute the probability-weighted sum of the

states with returns less than h, but the portfolio return and therefore the

ordering depends on both the magnitudes of the returns on the two assets in

each state and the portfolio weights. A simple numerical example is

sufficient to illustrate this point. Consider two states: one with a firm

return of �20% and a market return of �10% and the second with firm


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and market returns of �15%. For relatively larger (smaller) fractions in-

vested in the firm, the former will have a lower (higher) portfolio return than

does the latter, as illustrated in the table below.

State wi wm Ri Rm RP

1 0.6 0.4 �20% �10% �16%2 0.6 0.4 �15% �15% �15%1 0.4 0.6 �20% �10% �14%2 0.4 0.6 �15% �15% �15%

In spite of this dependence on the model parameters, there are some things

that can be said about the relevant measures of tail risk in this setting. First,holding all else fixed, the more extreme the negative returns on the firm, the

larger is the LPM of the portfolio, that is, the greater the tail risk. Because ofthe underdiversified nature of the portfolio, stock-specific risk matters. In the

context of tail risk, a natural measure of this stock-specific risk is the LPM of

the stock return:

LPMðRiÞ ¼X

Ri<hi

ðRi � hiÞ2: ð5Þ

Second, again holding all else fixed, the more extreme the negative returns

on themarket, the larger is theLPMof the portfolio. Therefore, in addition to

the stock-specific risk, a stock is risky to the extent that it contributes to thetail risk of the market portfolio. The natural measure of this component of

risk is the beta in the mean-lower partial moment CAPM setting:

�i;LPM ¼

XRm<hm

ðRi � hmÞðRm � hmÞ

XRm<hm

ðRm � hmÞ2

: ð6Þ

Third, and perhaps most interesting, is the risk component associated with

comovement of the firm and the market in the tail of the distribution of the

portfolio return. If the tail events for the firm and the market coincide, thenthese states will also be the tail states for the portfolio return, and the LPMof

the portfolio will be high. On the other hand, if they do not coincide, then weneed to develop an easily implementable empirical proxy for this comove-

ment. As noted above, the identity of the tail states for the portfolio depends

on the model parameters that determine the ordering of the returns acrossstates. State (j¼ 1, k¼ 1) is obviously the statewith the lowest portfolio return

independent of portfolio weights. The next lowest return state is either (j¼ 1,

k¼ 2) or (j¼ 2, k¼ 1) with returns

RP;12 ¼ wiRi;1 þ ð1� wiÞRm;2

RP;21 ¼ wiRi;2 þ ð1� wiÞRm;1:ð7Þ


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The former will have the lower return as long as

wiðRi;2 � Ri;1Þ > ð1� wiÞðRm;2 � Rm;1Þ: ð8Þ

This simple inequality generates some insight. Specifically, conditioning

on states with low firm returns (as opposed to low market returns), that

is, selecting (j¼ 1, k¼ 2) versus (j¼ 2, k¼ 1), is the intuitively correct

thing to do as long as the firm is more volatile than the market. The

difference between returns across discrete states is the analog to volatility,

and as long as the weight in the firm is sufficiently high, the set of low

portfolio return states will be those with low firm returns and varying

market returns rather than low market returns and varying firm returns.

This intuition motivates the construction of our hybrid measure of tail

risk, which we call hybrid tail covariance risk (H-TCR). Specifically, we

define

H-TCR ¼X

Ri<hi

ðRi � hiÞðRm � hmÞ: ð9Þ

The key distinction between this measure and the LPM beta in Equation

(6) is that H-TCR conditions on the states of the world with low stock

returns, not with low market returns. Note that for the purposes of

cross-sectional analyses, the denominator in Equation (6) is irrelevant

because it is equal across all stocks; it simply serves to normalize the

systematic risk measure.As a check on the possible economic impact of this distinction, we

perform a simple empirical exercise. For each month, six months or one

year of past daily returns (approximately 125 or 250 daily observations,

respectively) are used to determine the tail observations for the market

portfolio and also for individual stocks at the 10% level; that is, we iden-

tify the 13 or 25 days on which the market fell the most, and we also

separately identify the 13 or 25 days on which each individual stock fell

the most. We then count the number of days that these two sets have in

common for each individual stock. The table below shows percentiles for

the number of common days for the 1962–2012 sample period. For both

sample lengths, the median number of common days is small at 2.91 and

5.83 for six months and one year, respectively, an overlap of only approxi-

mately 20% between the tails of the market return distribution and that of

a typical stock in both cases. Even the 99th percentiles for the number of

common days are only 7.32 and 12.96 (an overlap of approximately 50%).

Clearly, the tail events for the market and individual stocks do not coin-

cide. In other words, tail events for individual stocks are primarily idio-

syncratic. Thus, there is a realistic possibility that H-TCR will differ

significantly from downside beta and moreover that this risk measure


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will better capture tail risk for investors with meaningful fractions of their

wealth in concentrated positions.

Percentiles for the number of common days in the 10% tail

1% 5% 10% 30% 50% 70% 90% 95% 99%

6 months 0.02 0.46 0.92 1.97 2.91 3.91 5.31 6.04 7.3212 months 0.68 1.79 2.45 4.30 5.83 7.39 9.73 10.89 12.96

We also look directly at the empirical, cross-sectional determinants of the

LPMof a concentrated portfolio of the type described in Equation (4). Again

using daily returns over a six-month or one-year period, the LPM of the

portfolio is calculated as

LPMðRpÞ ¼X

Rp<hp

ðRp � hpÞ2; ð10Þ

where the sum is taken over the days for which the portfolio return is less than

the specified threshold. Intuitively, this portfolio LPM will depend on the

three components of tail risk discussed above—systematic, stock-specific, and

hybrid.We consider two sets of portfolios weights—50% in the stock and 50% in

the market, and 30% in the stock and 70% in the market—and thresholds at

the 10th percentile of the relevant return distributions. Each month, we look

back over the preceding 6 or 12 months and calculate the four quantities in

Equations (5), (6), (9), and (10).We then run firm-level Fama-MacBeth cross-

sectional regressions ofLPMðRpÞ onLPMðRiÞ, �i;LPM , andH-TCRi for each

month from July 1963 to December 2012:

LPMpi;t ¼ �0;t þ �1;t LPMi;t þ �2;t �i;LPM;t þ �3;t H-TCRi;t þ "i;t: ð11Þ

For brevity, we only discuss the results for the one-year sample length, but the

results using six months are similar. For the 50/50 weights, the average slope

coefficient on LPMi is estimated to be 0.25 with a Newey-West t-statistic of

256.9, the average coefficient onH-TCRi is 0.37 with a t-statistic of 49.0, and

the average coefficient on �i;LPM is 0.0001 with a t-statistic of 2.2. All the

coefficients are statistically significant, but the magnitudes of the t-statistics

indicate their marginal explanatory power. Thus, although it is true that sys-

tematic tail riskmatters for the tail risk of concentrated portfolios, the relative

weakness of this effect suggests that it might be more difficult to detect in the

data. The average R-squared of the monthly cross-sectional regressions in

Equation (11) is 98.6%; that is, the three tail risk measures capture almost all

the cross-sectional variation in portfolio LPM.For the 30/70 weights, the average estimated slope coefficient on LPMi is

lower, 0.09, with a t-statistic of 134.1, because the investor allocates a smaller

amount to the individual stock. The average coefficient on H-TCRi is also


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lower at 0.26 with a t-statistic of 52.1, whereas that on �i;LPM is higher at0.0002, with a corresponding t-statistic of 2.7. The average R-squared of themonthly cross-sectional regressions is 94.8%. In both cases, the significance ofthe coefficient on H-TCR and the high explanatory power of the cross-sec-tional regressions validate our choice of the hybrid tail risk measure.Moreover, although variations of the H-TCR measure give similar results,these variations generally yield lower cross-sectional R-squareds than thosefor H-TCR.

Overall, these empirical results indicate that H-TCR is an appropriatemeasure of risk in our framework. Theoretically, the dependence ofH-TCR on the colower partial moment of firm and market returns followsfrom the assumption that the LPM of the portfolio return is the correctmeasure of risk at the portfolio level. Finally, as is true with any modelthat assumes concentrated holdings, the total risk of these individual assetswill also contribute to portfolio risk and therefore may require compensationin equilibrium.

2. Data and Variable Definitions

The first dataset includes all New York Stock Exchange (NYSE), AmericanStock Exchange (AMEX), and NASDAQ financial and nonfinancial firmsfrom the Center for Research in Security Prices (CRSP) for the period fromJuly 1962 through December 2012.8 We use daily stock returns to estimatealternative measures of risk. The second dataset is COMPUSTAT, which isprimarily used to obtain the book values for individual stocks.

For each month from July 1963 to December 2012, we compute the threetail risk measures for each firm in the sample—(1) the LPM of the return onthe stock, (2) the LPM beta of the stock with respect to the market, and(3) hybrid tail covariance risk, as defined in Equations (5), (6), and (9),respectively. In all cases we use daily returns over the past six months,except for certain extensions in Section 4, and the return thresholds for thestock and market return are determined by the relevant empirical percentilesover the same sample. Formuch of the analysis, we employ the 10th percentileas our measure of the tail of the distribution, but we also report results forthresholds ranging from the 5th percentile to the 50th percentile.

We also employ an extensive set of control variables. As Subrahmanyam(2010) points out, over fifty variables have been shown to have predictivepower for stock returns in the cross-section. It is infeasible to control for all of

8 Following Harris (1994), Jegadeesh and Titman (2001), and Ang, Chen, and Xing (2006), we remove small,illiquid, and low-priced stocks from the sample. Specifically, for each month, all NYSE stocks on CRSP aresorted by firm size to determine the NYSE decile breakpoints for market capitalization. Then we exclude allNYSE/AMEX/NASDAQ stocks with market capitalizations that would place them in the smallest NYSE sizedecile. We also exclude stocks whose price is less than $5 per share. This filter is important for reducing liquidityand other microstructure biases (Asparouhova, Bessembinder, and Kalcheva 2010, 2013). However, we alsoconduct the firm-level cross-sectional regressions on the full sample as discussed in Section 4.


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these variables, but we select both the most popular variables in the literatureand those that, intuitively, are most likely to be correlated with our tail riskmeasures. The first four variables are the widely used cross-sectional returnpredictors—market beta, size, book-to-market, and momentum. We alsocontrol for net share issuance and microstructure-related phenomena in theform of short-term reversals and liquidity. Finally, we include four vari-ables—coskewness, downside beta, idiosyncratic volatility, and extremepositive returns—that are directly related to the distribution of returns, andthus possibly tail risk as well, and that have been shown to have significantpredictive power. The detailed definitions of all these variables are provided inthe Appendix.

3. Preliminary Evidence

Given the number of potential control variables, that is, other stock charac-teristics that may influence returns, the Fama-MacBeth cross-sectional re-gression approach may be the natural way to examine the predictive powerof measures of tail risk.We turn to these regressions in Section 4; however, toget an initial feel for the data, we first look at univariate sorts on the basis ofour three tail risk measures and the associated characteristics of theportfolios.

3.1 Average returns for univariate portfolio sorts

Table 1 presents the average monthly returns for the equal-weighted andvalue-weighted decile portfolios that are formed by sorting the NYSE,AMEX, and NASDAQ stocks based on our three tail risk measures—H-TCR, LPM(Ri), and �LPM. We also report the average across months ofthe median tail risk measure within each portfolio. The returns are reportedfor the sample period July 1963 to December 2012, whereas the measures oftail risk are computed over the preceding six months. Portfolio 1 (Low) con-tains stocks with the lowest tail risk and Portfolio 10 (High) includes stockswith the highest tail risk in the previous six months.We turn first to our new hybrid measure of tail covariance risk. By con-

struction, the average H-TCR of individual stocks in the univariate sort in-creases monotonically across the deciles, from�0.262 for Portfolio 1 to 0.130for Portfolio 10. Becausewe are conditioning on states inwhich the individualstock return is less than the specified threshold, the stock specific term inEquation (9) is always negative. Thus, a negative (positive) H-TCR indicatesthat the market term is positive (negative), on average, in these same states.Positive and large H-TCRs correspond to stocks whose low returns coincidewith those of themarket as a whole. In other words, they have substantial tailrisk because a portfolio with significant weights in both the stock and themarket will tend to have returns in the left tail due to the coincidence of tailevents for both assets.


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As shown in the second column, the equal-weighted average return of

individual stocks is about 0.48% per month for the low H-TCR decile

(Portfolio 1) and 1.22% per month for the high H-TCR decile (Portfolio

10). The raw average return difference between deciles 10 and 1 is 0.74%

per month (8.9% per annum) with a Newey-West (1987) t-statistic of 5.40.9

The value-weighted return difference is smaller (0.39% per month), but it is

still statistically significant. In other words, there is evidence that our hybrid

measure of tail risk is priced in the cross-section consistent with the model in

Section 1. However, there is also some evidence of nonmonotonicity in the

average portfolio returns, and we have not yet made an effort to control for

other priced risks that may vary across these portfolios. We do so in the firm-

level cross-sectional regressions in Section 4.The results for the other two tail risk measures are in sharp contrast to

those for H-TCR. When stocks are sorted on the LPM of their daily returns

over sixmonths, thismeasure of stock-specific tail risk is negatively associated

with raw portfolio returns. That is, the average returns on stocks with high

LPMs are lower than those with less risk, with return differences of�0.69%

and �0.62% for equal- and value-weighted portfolios, respectively, that is

Table 1

Univariate portfolios of stocks sorted by tail risk measures

TCR LPM(Ri) �LPM

H-TCR EWreturn

VWreturn

LPM(Ri) EWreturn

VWreturn

�LPM EWreturn

VWreturn

Low �0.262 0.48 0.46 0.070 1.08 0.89 �1.347 0.91 0.742 �0.147 0.86 0.55 0.140 1.17 0.99 �0.633 1.07 0.903 �0.098 0.96 0.79 0.200 1.18 0.95 �0.194 1.10 0.904 �0.064 1.06 0.90 0.270 1.24 0.89 0.191 1.15 0.955 �0.037 1.16 0.80 0.350 1.22 1.06 0.568 1.10 0.916 �0.013 1.14 0.96 0.460 1.22 0.95 0.962 1.20 0.957 0.011 1.15 0.92 0.600 1.14 0.89 1.409 1.16 0.908 0.037 1.30 0.90 0.830 1.08 0.78 1.959 1.17 1.019 0.070 1.27 1.01 1.230 0.88 0.79 2.726 0.99 0.81High 0.130 1.22 0.85 2.530 0.39 0.27 4.165 0.75 0.77

Return diff. 0.74 0.39 �0.69 �0.62 �0.15 0.03t-stat. (5.40) (2.01) (�2.88) (�2.61) (�0.53) (0.11)

Decile portfolios are formed every month from July 1963 to December 2012 by sorting stocks based on threemeasures of tail risk over the past sixmonths: (1) hybrid tail covariance risk (H-TCR,Equation (9)), (2) the lowerpartial moment of returns (LPM(Ri), Equation (5)), and (3) LPM beta with respect to the market (�LPM,Equation (6)). Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) tail risk. We report theequal-weighted (EW) and value-weighted (VW) average monthly returns (in percentage terms) and the averagetail riskmeasure for each portfolio. The last two rows present the differences in averagemonthly returns betweenportfolios 10 and 1 and the associated Newey-West (1987) adjusted t-statistics, in parentheses.

9 We also replicate this analysis for the 1926–1962 sample period. For brevity, these results are not reported, butboth the raw average return difference and the associated four-factor alpha are small and statistically insignifi-cant. There are a number of possible explanations for this negative result, but one explanation that is consistentwith the motivation in Section 1, is that investors did not hold concentrated portfolios that also containedpositions in well-diversified portfolios during this earlier period when mutual funds and defined contributionretirement plans were scarcer.


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economically large in magnitude and statistically significant (with t-statisticsof �2.88 and �2.61). Although this result is somewhat disappointing fromthe perspective of uncovering priced tail risk in our framework of under-diversified holdings, it is perhaps not totally surprising. As we analyze inmore detail below, LPM is correlated with other measures of stock-specificrisk, specifically volatility (Ang et al. 2006, 2009) and extreme returns (Bali,Cakici, and Whitelaw 2011), that have been shown to have a strong relationto returns in the cross-section. Thus, isolating the effect of stock-specific tailrisk may be extremely difficult.Finally, ourmeasure of systematic tail risk, �LPM, is very weakly associated

with portfolio returns in the cross-section. The sign of the difference betweenthe returns on Portfolios 10 and 1 depends on the weighting scheme, thesedifferences are economically and statistically insignificant, and the portfolioreturns are clearly nonmonotonic. In light of the voluminous literature at-tempting, and in many cases failing, to find significant pricing of systematicrisk measures in the cross-section, this result is not totally unexpected.Moreover, systematic tail risk is the least important of the three tail riskmeasures in determining the tail risk of concentrated portfolios, as shownin Section 1.

3.2 Descriptive statistics for tail risk portfolios

While the raw return differences between the high and lowH-TCRdeciles areeconomically and statistically significant, the pattern across deciles in rawreturns is not quitemonotonic.Moreover, stock-specific tail risk, asmeasuredby LPM, appears to be negatively priced in raw returns. These patterns in thedata could be the result of additional priced risk factors, and these factorsmight also influence the risk-adjusted return differences across portfolios. Tohighlight the firm characteristics and risk attributes of stocks in the portfoliosof Table 1, Table 2 presents descriptive statistics for the stocks in the variousdeciles.As a first pass at understanding the interaction of tail risk with firm

characteristics and risk attributes, we compute the firm-level cross-sectionalcorrelations of the three tail risk measures and a variety of other variables—downside beta (�down), the price (in dollars), the market beta, the log marketcapitalization (in millions of dollars), the book-to-market (BM) ratio, thereturn over the 6 months prior to portfolio formation (MOM), the returnin the portfolio formation month (REV), a measure of illiquidity (scaled by105), the coskewness, the idiosyncratic volatility, themaximumdaily return inthe portfolio formationmonth (MAX), and the net share issuance (definitionsof these variables are given in theAppendix)—for eachmonth from July 1963to December 2012. Table 2, panel A, reports the time-series averages of thesecross-sectional correlations. Hybrid tail risk, H-TCR, is positively correlatedwith market beta, downside beta, size, and momentum, and negatively


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Table

2

Descriptive

statisticsfordecileportfoliosofstockssorted

bytailrisk

measures

H-TCR

LPM(R

i)�LPM

�down

Price

BETA

SIZ

EBM

MOM

REV

ILLIQ

COSKEW

IVOL

MAX

ISSUE

Panel

A:Averagefirm

-level

correlations

H-TCR

1�0.421

0.361

0.214

0.108

0.233

0.279

0.013

0.133

0.026�0.159

�0.141

�0.298�0.161�0.001

LPM(R

i)1

0.303

0.295�0.091

0.264�0.225�0.057�0.095�0.051

0.059

�0.038

0.443

0.295�0.085

�LPM

10.801�0.049

0.758�0.060�0.086�0.027�0.059�0.062

�0.262

0.324

0.282�0.139

�down

1�0.049

1.008�0.028�0.126�0.132�0.077�0.104

�0.156

0.363

0.316�0.144

Price

1�0.022

0.336�0.096

0.069

0.031�0.162

0.028

�0.172�0.119�0.014

BETA

10.001�0.125

0.014�0.006�0.122

0.027

0.369

0.346�0.173

SIZ

E1

�0.149�0.012�0.013�0.438

0.046

�0.415�0.303

0.060

BM

10.061

0.030

0.117

�0.009

�0.041�0.022

0.070

MOM

10.009�0.050

�0.068

0.033

0.016�0.070

REV

10.006

�0.004

0.186

0.425�0.017

ILLIQ

10.000

0.202

0.136�0.003

COSKEW

1�0.022

0.001

0.034

IVOL

10.903�0.121

MAX

1�0.102

ISSUE

1Panel

B:H-TCR

Low

H-TCR

�0.24

1.37

0.53

0.89

17.47

0.84

5.19

0.59

�1.95

0.12

12.55

�0.06

2.59

5.71

�4.52

2�0.13

0.58

0.41

0.82

20.79

0.81

5.49

0.64

6.65

0.74

10.29

�0.06

2.13

4.83

�3.95

3�0.08

0.40

0.40

0.80

22.99

0.79

5.71

0.66

8.96

0.87

7.90

�0.07

1.92

4.41

�3.52

4�0.05

0.32

0.44

0.80

24.43

0.80

5.87

0.68

10.38

1.03

6.59

�0.08

1.79

4.18

�3.28

5�0.03

0.27

0.50

0.82

25.97

0.81

6.03

0.69

11.35

1.08

5.44

�0.08

1.68

3.99

�3.10

60.00

0.25

0.61

0.84

27.52

0.84

6.20

0.69

12.20

1.08

4.46

�0.09

1.59

3.85

�2.97

70.02

0.25

0.76

0.88

28.76

0.88

6.36

0.69

13.12

1.12

3.67

�0.11

1.54

3.79

�3.06

80.04

0.28

1.03

0.96

30.04

0.96

6.50

0.69

13.95

1.15

3.10

�0.12

1.52

3.80

�3.20

90.08

0.34

1.47

1.07

31.10

1.07

6.62

0.67

16.07

1.19

2.76

�0.14

1.53

3.96

�3.80

HighH-TCR

0.13

0.61

2.68

1.35

30.20

1.35

6.53

0.62

20.94

1.08

3.09

�0.17

1.76

4.63

�5.29

(continued)


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Table

2Continued

H-TCR

LPM(R

i)�LPM

�down

Price

BETA

SIZ

EBM

MOM

REV

ILLIQ

COSKEW

IVOL

MAX

ISSUE

Panel

C:LPM(R

i)

Low

LPM(R

i)�0.012

0.08

�0.09

0.57

32.86

0.56

6.85

0.79

11.30

1.13

2.64

�0.07

1.10

2.65

�1.31

2�0.012

0.14

0.28

0.71

31.61

0.71

6.67

0.72

12.02

1.16

3.05

�0.08

1.30

3.16

�2.08

3�0.010

0.19

0.51

0.79

30.08

0.78

6.49

0.70

12.34

1.20

3.52

�0.09

1.44

3.50

�2.62

4�0.010

0.26

0.70

0.85

28.45

0.85

6.30

0.68

12.56

1.17

4.25

�0.09

1.57

3.81

�3.16

5�0.010

0.33

0.88

0.91

26.73

0.90

6.10

0.67

12.77

1.10

4.96

�0.10

1.71

4.13

�3.51

6�0.012

0.43

1.08

0.97

25.14

0.96

5.93

0.65

13.30

1.15

5.68

�0.10

1.86

4.47

�3.98

7�0.015

0.56

1.26

1.03

23.43

1.02

5.75

0.64

13.48

1.06

6.55

�0.11

2.03

4.86

�4.50

8�0.018

0.76

1.52

1.11

21.84

1.09

5.60

0.62

12.90

0.91

7.45

�0.11

2.24

5.32

�5.25

9�0.031

1.12

1.77

1.20

20.11

1.18

5.42

0.59

11.23

0.69

8.66

�0.11

2.48

5.83

�6.02

HighLPM(R

i)�0.098

2.33

2.11

1.34

17.16

1.29

5.20

0.55

1.43

�0.48

9.80

�0.10

2.88

6.53

�6.63

Panel

D:�LPM

Low�LPM

�0.085

0.24

�1.17

0.41

25.29

0.44

5.77

0.75

12.95

1.79

8.82

�0.02

1.58

3.55

�1.88

2�0.054

0.21

�0.48

0.55

27.35

0.55

6.09

0.74

11.07

1.29

6.01

�0.06

1.41

3.26

�1.97

3�0.040

0.23

�0.05

0.66

27.72

0.65

6.19

0.71

10.75

1.15

4.97

�0.07

1.47

3.44

�2.38

4�0.028

0.26

0.31

0.76

27.70

0.75

6.22

0.69

10.48

1.05

4.45

�0.08

1.53

3.64

�2.81

5�0.017

0.30

0.67

0.86

27.56

0.84

6.23

0.67

10.79

0.94

4.26

�0.09

1.61

3.85

�3.19

6�0.004

0.34

1.04

0.95

26.90

0.94

6.20

0.65

10.63

0.93

4.37

�0.10

1.69

4.10

�3.52

70.009

0.40

1.45

1.06

26.24

1.05

6.14

0.63

11.40

0.81

4.55

�0.12

1.80

4.39

�4.11

80.025

0.49

1.96

1.21

25.02

1.19

6.04

0.61

12.30

0.71

4.84

�0.13

1.96

4.81

�4.90

90.045

0.64

2.67

1.40

23.34

1.38

5.86

0.59

13.88

0.54

5.39

�0.13

2.19

5.48

�6.04

High�LPM

0.090

1.05

4.02

1.75

19.75

1.71

5.55

0.55

17.83

0.15

6.62

�0.15

2.64

6.67

�8.74

Decile

portfolio

sareform

edeverymonth

from

July1963

toDecem

ber

2012

bysortingstocksbased

onthreemeasuresoftailrisk

overthepastsixmonths:(1)hybridtailcovariance

risk

(H-

TCR)(Equation(9)),(2)

thelower

partialmomentofreturns(LPM(R

i),Equation(5)),and(3)LPM

betawithrespectto

themarket(�

LPM,E

quation(6)).P

ortfolio

1(10)

istheportfolio

of

stockswiththelowest(highest)tailrisk.P

anelApresentsthetime-seriesaveragesofthecross-sectionalcorrelationsofallthevariables.PanelBreportsforeach

H-TCRdecile

theaverageacross

themonthsinthesampleofthemedianvalueswithineach

month

ofvariouscharacteristicsforthestocks—

thethreetailrisk

measures,downsidebeta,theprice

(indollars),themarketbeta,the

logmarketcapitalization,thebook-to-m

arket(BM)ratio,thecumulative

return

overthe6monthspriorto

portfolioform

ation(lab

eled

MOM),thereturn

intheportfolioform

ationmonth

(labeled

REV),ameasureofilliquidity(scaledby10

5),thecoskew

ness(C

OSKEW),idiosyncraticvolatility

(IVOL),themaxim

um

dailyreturn

overthepastonemonth

(MAX),andnetshare

issuance.PanelsCandD

presentthesamedescriptive

statistics

fordecile

portfolio

sofLPM(R

i)and�LPM,respectively.


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correlated with illiquidity, coskewness, idiosyncratic volatility, and MAX.

The correlations of H-TCR with book-to-market, reversals, and net share

issuance are very small, with magnitudes less than 0.03. Stock-specific tail

risk, LPM(Ri), is strongly positively correlated with both idiosyncratic vola-

tility and MAX, as is systematic tail risk, �LPM.Although these correlations are useful summaries of the possible inter-

actions across variables, they cannot reveal nonmonotonicities or other fea-

tures of the data that may affect the results. To examine these relations in

more detail, panels B through D report the average across the months in the

sample of the median values within each month of various characteristics for

the stocks in each decile sorted byH-TCR, LPM(Ri), and �LPM, respectively.

In each case, we report values for the three tail risk measures and the twelve

other variables.Table 2, panel B, reports the characteristics for the portfolios sorted on

H-TCR.10 Our hybrid measure of tail risk is positively related to systematic

tail risk, asmeasured by�LPM, but nonmonotonically related to stock-specific

tail risk, as measured by LPM(Ri). This latter result is a manifestation of the

fact that many tail events for individual stocks are idiosyncratic. Stocks with

large idiosyncratic negative returns have high values of LPM but low values

of H-TCR, whereas stocks with large systematic negative returns have high

values of both LPM and H-TCR.Interestingly, stocks with high H-TCR are larger, higher priced, and more

liquid stocks, on average. The intuition behind this result is that while smaller

stocks tend to have more extreme negative returns, these tail events are also

more likely to be idiosyncratic. Thus, in the context of our hypothesized

portfolios of concentrated positions in individual stocks plus additional

wealth in a well-diversified fund, it is the larger stocks that generate more

portfolio tail risk after controlling for the stock-specific component. This size

and liquidity discrepancy suggests that the raw return difference will hold up

to risk adjustment on these dimensions. Large stocks and liquid stocks, on

average, have low returns, whereas stocks with low systematic risk in the left

tail (low H-TCR) are small, illiquid stocks that should have high returns, all

else equal. Apparently, in the raw returns, the effect of hybrid tail risk dom-

inates the effect of size or liquidity on future returns.A second important implication of these characteristics is that the liquidity

and related microstructure biases identified by Asparouhova, Bessembinder,

and Kalcheva (2010, 2013) are of little or no concern in the context of H-

TCR. Specifically, these papers show that the measured returns on small,

illiquid, and low-priced stocks tend to be biased upward because of micro-

structure effects inherent in the prices of these securities. We have excluded

10 The average acrossmonths of themedianH-TCR for each portfolio differs slightly from that reported in Table 1because the sample is slightly smaller due to the data requirements necessary to calculate some of the othervariables.


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the most extreme of these stocks with our data filters described in Section 2,but any residual effect will work against finding a positive return associatedwith H-TCR due to the characteristics of the high H-TCR portfolios.Both market beta and downside beta also increase as H-TCR increases,

implying that stocks with high hybrid covariance risk in the lower tail of thestock return distribution aremore exposed tomarket risk as well as downsiderisk. Of course, systematic total and downside risk likely do not explain theraw return differences across portfolios in Table 1 because market beta anddownside beta are weakly priced, at best, in the cross-section of future stockreturns.In contrast, it will be important to control for momentum when risk-ad-

justing returns. Stocks with high H-TCR (low H-TCR) are generally pastwinners (losers) over a horizon of 6 months, and thus H-TCR-sorted port-folios should exhibit the well-documented intermediate-term momentumphenomenon. On the other hand, both median book-to-market ratios(BM) and average returns in the portfolio formationmonth (REV) are similaracross the H-TCR portfolios, indicating no association between H-TCR andthe value premium or short-term reversals.COSKEW (Harvey and Siddique 2000) measures the direction and

strength of the relation between individual stock returns and squaredmarket returns. A preference for positive skewness suggests a negative pricefor coskewness risk. Panel B indicates that stocks with highH-TCR also havelow coskewness, indicating that it will be important to control for thisphenomenon.Panel B also examines two properties of the stock return distribution—

idiosyncratic volatility and the prevalence of extreme positive returns—bothof which have been linked to expected returns in the literature. Stocks withhigh H-TCR seem to have somewhat lower idiosyncratic volatility and lowermaximum daily returns in the portfolio formation month. Interestingly, thepatterns across portfolios in both IVOL and MAX do superficially resemblethose in the raw returns in Table 1.The final column of Panel B examines the interaction betweenH-TCR and

net share issues (ISSUE). As described in theAppendix, wemeasure net shareissues via CRSP, using the change over 12 months in split-adjusted sharesoutstanding, which is negative when firms on balance repurchase during the12-month period and positive when on balance they issue. Panel B shows thatalthough there is no monotonic pattern of ISSUE when moving from lowH-TCR to highH-TCRportfolios, compared to firms with lowH-TCR, highH-TCR firms, on average, issue less new equity and/or repurchase moreequity. Because earlier studies find a strong negative relation between netshare issues and future returns, it is important to control for ISSUE.Panel C reports the same characteristics as panel B for portfolios sorted on

LPM(Ri) rather than on H-TCR. These characteristics may suggest a poten-tial explanation for the anomalous negative relation between raw returns and


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stock-specific tail risk in Table 1. Clearly, such an explanation cannot rely onmarket beta, downside beta, size, or illiquidity, because these effects go in theopposite direction to the raw returns across the deciles. The stock-specificreturn distribution measures idiosyncratic volatility (IVOL) and extremepositive returns (MAX) are more likely candidates. Both of these variableshave a strong negative relation to returns in the cross-section and increasemonotonically across the LPM(Ri)-sorted portfolios. This association be-tween LPM, idiosyncratic volatility, and extreme returns is both expectedand probably difficult to resolve empirically.

In panel D, we report the characteristics of portfolios sorted on our finaltail risk measure, systematic tail risk as measured by �LPM. There is little ornothing surprising in the results. Tail beta is positively associated withmarketbeta, downside beta, coskewness, idiosyncratic volatility, and extreme posi-tive returns.

4. Firm-Level Cross-Sectional Regressions

The univariate-sort portfolio results in Table 1 are certainly consistent withH-TCR being priced in the cross-section, while the evidence for stock-specificand systematic tail risk is negative, but Table 2 identifies a number of riskfactors and firm characteristics that may play a role in the results. Therefore,we now examine the cross-sectional relation between tail risk and expectedreturns at the firm level using the Fama and MacBeth (1973) methodology.Specifically, we run the following multivariate specification and nested ver-sions thereof:

Ri;tþ1 ¼ �0;t þ �1;tXi;t þ �2;tBETAi;t þ �3;tSIZEi;t þ �4;tBMi;t

þ �5;tMOMi;t þ �6;tZi;t þ "i;tþ1;ð12Þ

where Xi,t are the three tail risk measures—H-TCR, LPM(Ri), and �LPM;BETA, SIZE, BM, and MOM are the four standard control variables; andZi,t represents the possible inclusion of other control variables.

The motivation for this analysis indicates that it is the tail risk of concen-trated portfolios that may be priced, and the empirical evidence in Section 1shows that all three tail risk measures contribute to this risk in the cross-section. Consequently, we focus on regressions with all three measuresincluded as separate variables, although for robustness we also report someresults with the measures included individually. One might also argue forcombining hybrid, systematic, and stock-specific tail risk into a single meas-ure, but, as shown in Section 1, their relative contributions to portfolio riskdepend critically on the precise nature of the underdiversification in the port-folio. Therefore, we allow the coefficients to vary across the variables.

Table 3, panel A, reports the time-series averages of the slope coefficientsfor the sample period July 1963–December 2012 (594 monthly observations)


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Table 3

Firm-level cross-sectional return regressions

Panel A: Specifications with standard control variables

Tail risk measures Standard control variables

H-TCR LPM(Ri) �LPM BETA SIZE BM MOM MOM12

2.5300

(4.29)

�0.0510

(�0.51)

0.0100

(0.18)

1.6724 �0.0510 0.0317

(2.36) (�0.74) (0.37)

1.9996 �0.1419 �0.1079 0.1164 0.7662

(5.11) (�0.84) (�2.68) (1.83) (3.85)

�0.1759 �0.0264 �0.0990 0.1209 0.8020

(�4.07) (�0.16) (�2.36) (1.89) (3.97)

�0.0205 �0.0738 �0.0752 0.1379 0.8239

(�0.62) (�0.48) (�1.77) (2.15) (4.08)

2.1077 �0.0695 �0.0276 �0.0623 �0.1203 0.1024 0.7495

(4.56) (�1.19) (�0.49) (�0.41) (�3.08) (1.62) (3.83)

1.2932 �0.1060 �0.1660 0.1854 0.6972

(3.43) (�0.68) (�3.26) (3.04) (4.69)

�0.0858 �0.0579 �0.1537 0.1909 0.7340

(�2.28) (�0.40) (�2.77) (3.08) (4.82)

0.0019 �0.0814 �0.1431 0.2005 0.7174

(0.07) (�0.56) (�2.47) (3.22) (4.56)

1.9262 �0.0839 �0.0131 �0.0872 �0.1217 0.1203 0.7779

(4.26) (�1.48) (�0.20) (�0.58) (�3.08) (1.98) (5.21)

Panel B: Specifications with standard and additional control variables

Tail risk measures Standard control variables Additional control variables

H-TCRLPM(Ri) �LPM BETA SIZE BM MOM �down REV ILLIQ COSKEW IVOL MAX ISSUE

2.1077 �0.0695 �0.0276 �0.0623 �0.1203 0.1024 0.7495

(4.56) (�1.19) (�0.49) (�0.41) (�3.08) (1.62) (3.83)

1.9014 �0.0712 �0.0333 0.0546 �0.1241 0.1060 0.7380 �0.0703

(4.08) (�1.24) (�0.76) (0.20) (�3.21) (1.69) (4.15) (�0.23)

2.3927 �0.0734 �0.0675 �0.0009 �0.1281 0.1160 0.6755 �3.8955

(4.81) (�1.16) (�1.46) (�0.01) (�3.24) (1.77) (3.30) (�9.18)

2.0583 �0.0705 �0.0219 �0.0790 �0.1465 0.1036 0.7376 �0.0027

(4.45) (�1.22) (�0.37) (�0.52) (�3.69) (1.64) (3.74) (�0.64)

2.0343 �0.0645 �0.0667 �0.0207 �0.1306 0.0963 0.7722 �0.1970

(4.41) (�1.15) (�1.26) (�0.12) (�3.40) (1.54) (4.02) (�0.87)

1.0860 �0.0237 0.0354 0.0631 �0.1803 0.0821 0.7997 �0.2865

(2.41) (�0.41) (0.53) (0.43) (�4.80) (1.32) (4.10) (�9.21)

1.4082 �0.0466 0.0070 0.0840 �0.1638 0.0917 0.7479 �8.5610

(3.11) (�0.78) (0.11) (0.57) (�4.26) (1.46) (3.81) (�9.76)

2.1192 �0.0748 �0.0285 �0.0860 �0.1188 0.1134 0.7188 �0.4651

(4.65) (�1.31) (�0.51) (�0.57) (�3.04) (1.81) (3.69) (�4.22)

1.3392 �0.0419 �0.0185 1.7642 �0.2003 0.1135 0.1790 �1.8165 �4.9176 0.0001 �1.7782 �0.2461 �2.8657 �0.4443

(2.66) (�0.68) (�0.35) (4.51) (�5.48) (1.80) (0.96) (�4.39) (�11.88) (0.02) (�5.61) (�4.67) (�3.54) (�4.20)

For each month from July 1963 to December 2012, we run a firm-level cross-sectional regression of the one-month-ahead return on lagged predictor variables including our tail risk measures—H-TCR, LPM(Ri), and�LPM—the standard control variables (in panel A), and seven additional control variables (in panel B). Thevariables are defined in the Appendix. Two alternative definitions of momentum are used;MOM is the 6-monthcumulative return from month t–7 to month t–2, and MOM12 is the 11-month cumulative return from montht–12 to month t–2. In each row, panels A and B report the time-series averages of the cross-sectional regressionslope coefficients and their associated Newey-West (1987) adjusted t-statistics (in parentheses).


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from regressions of one-month-ahead stock returns on combinations of our

three tail risk measures and the four standard controls.11 The average slopes

provide standard Fama-MacBeth tests for determining which explanatory

variables on average have nonzero premiums, and the Newey-West adjusted

t-statistics are given in parentheses.Not surprisingly, the univariate cross-sectional regression results are

broadly consistent with the raw return differences across portfolios from

the univariate portfolio sorts in Table 1. The average slope on H-TCR is

2.53 with a t-statistic of 4.29. Given a difference in median H-TCR of ap-

proximately 0.39 between the high and low H-TCR deciles, this coefficient

estimate translates into amonthly return difference of 1%, about 40%higher

than the economic effect seen in Table 1 for the equal-weighted portfolios.

For both LPM(Ri) and �LPM, the coefficients are economically small and

statistically insignificant. When all three tail risk measures are included to-

gether, the results are qualitatively similar, although the coefficient on

H-TCR falls to 1.67 with a t-statistic of 2.36, yielding an estimated monthly

return difference of 0.65% between high and low H-TCR stocks.In the regressions including the standard control variables, the coefficients

on these variables are as expected. The average slope onBETA is negative and

statistically insignificant, which is consistent with prior empirical evidence.

The average slope on SIZE is negative and significant; the average slope on

BM is positive and significant inmost specifications; and stocks exhibit strong

intermediate-term momentum, whether proxied by the cumulative return

over the 6 months (MOM) or 12 months (MOM12) prior to the return pre-

diction month.Of greater interest are the average slope coefficients on the tail risk meas-

ures. The predictive power ofH-TCR remains significant after accounting for

the standard controls, with both alternative definitions of momentum, and

after including the other tail risk measures. In the most complete specifica-

tions, the coefficient on H-TCR is about two with t-statistics greater than

four. The implied economic magnitude of the effect is slightly lower than in

the univariate regression and slightly larger than in the regression that only

includes tail risk measures, but at a difference of approximately 75 to 95 basis

points per month between median stocks in the high and lowH-TCR deciles,

it is still large. The slight attenuation is due partly to the inclusion of the

momentum variable, with which H-TCR is positively correlated. The coeffi-

cient in a similar regression controlling only for the market beta, size, and

book-to-market is larger.

11 The results reported in Table 3 and elsewhere are for the sample that excludes small, illiquid, and low-pricedstocks as described in Section 2. However, we also replicate Table 3 for the full sample. For brevity, these resultsare not reported, but the qualitative results on this larger sample are similar, albeit being slightly weaker, perhapsbecause of the microstructure biases identified by Asparouhova, Bessembinder, and Kalcheva (2010, 2013) anddiscussed in Section 3 or simply because of the additional noise in the data.


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Controlling for the additional risk factors has a large effect on the LPM(Ri)

coefficient when it is the only tail risk measure, but, in the more complete

specifications, themagnitudemoves closer to that in the univariate regression.

For all specifications, the coefficient is negative, and it is sporadically statis-

tically significant. However, the sign on the coefficient is inconsistent with

LPM(Ri) being a measure of priced tail risk—stocks with high stock-specific

tail risk apparently have lower expected returns. This anomalous result

strongly suggests that this variable is proxying for an omitted factor.For systematic tail risk, the inclusion of the additional factors has little

effect. The coefficient remains small in magnitude and of the wrong sign in

regressions with the standard control variables included, albeit it is statistic-

ally insignificant in all cases.Table 3, panel B, reports results formultivariate regressions that include all

three tail risk measures, the standard control variables, and each of the add-

itional control variables from Table 2 in turn. The coefficients on these add-

itional variables are generally in line with the existing literature. The average

slope on REV is negative and highly significant, implying that stocks exhibit

strong short-term reversals. There is no evidence of a significant link between

expected returns and illiquidity, coskewness, and downside beta. Consistent

with the findings of Ang et al. (2006) and Bali, Cakici, and Whitelaw (2011),

the results indicate a negative and significant relation between expected re-

turns and the IVOL and MAX variables. Finally, in line with Pontiff and

Woodgate (2008) and Fama and French (2008), we find a strong negative

relation between net share issues and future stock returns.12

Again, it is the tail risk variables that are of primary interest. For H-TCR,

the inclusion of controls for downside beta, return reversals, illiquidity, cos-

kewness, or net share issues has little effect on the coefficient. However, both

IVOL and MAX reduce the coefficient on H-TCR by approximately 50%.

The effects of these control variables are extremely strong in the data with a

sign opposite to that of H-TCR, and they are negatively correlated with

H-TCR. Nevertheless, H-TCR remains economically and statistically signifi-

cant in all the specifications.Interestingly, the pattern of coefficients across regressions is similar for

LPM(Ri). Downside beta, reversals, illiquidity, coskewness, and net share

issues have little effect, but the inclusion of either IVOL or MAX again

reduces the coefficient by about 50%. In this case, the sign of the coefficient

on LPM(Ri) and the control variables (IVOL, MAX) is the same, but the

variables are positively correlated. Regardless of the specification, the

12 The new issues puzzle refers to the result that stocks of firms that issue new equity are, on average, very poorinvestments relative to various benchmarks. In contrast, firms that repurchase equity significantly outperformbenchmark returns (see, e.g., Loughran and Ritter 1995; Spiess and Affleck-Graves 1995; and Ikenberry,Lakonishok, and Vermaelen 1995). Consistent with the findings of other studies on share issue-repurchase(e.g., Daniel and Titman 2006; McLean, Pontiff, and Watanabe 2009; and Bali, Demirtas, and Hovakimian2010), we find a negative link between net share issuance and the cross-section of expected returns.


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anomalous sign on stock-specific tail risk is preserved, but the coefficient is

not statistically significant.For systematic tail risk, it is the control for coskewness that has the largest

effect on the results, but, as with stock-specific tail risk, none of the coeffi-

cients is statistically significant. Of minor interest, including IVOL or MAX

does cause the sign to flip from its theoretically anomalous negative value to a

positive value.Finally, the last row includes all three measures of tail risk as well as the

standard and additional control variables simultaneously. Given the number

of variables and their complex correlation structure, the results need to be

interpreted with caution because the small sample properties are uncertain.

However, similar to our earlier findings, H-TCR positively predicts future

returns, whereas the other tail risk measures, LPM(Ri) and �LPM, have no

predictive power. Interestingly, we find a positive and significant market risk

premium and negative and significant premiums on downside beta and

coskewness.The tail risk measures in Tables 1, 2, and 3 are constructed using the 10%

tails of the relevant return distributions over the preceding six months. The

choice of 10% and six months is somewhat arbitrary, although it intuitively

provides a reasonable trade-off between a sufficient number of observations

to limit estimation error and the desire to get ameasure of tail risk rather than

more general downside risk. We now investigate the predictive power of al-

ternative measures of H-TCR, LPM(Ri), and �LPM estimated using different

tails of the daily return distribution over the preceding year (instead of six

months).13 Moreover, there is the more fundamental issue of the nature of

investor preferences or the asymmetry properties of joint return distributions

that generate a role for tail risk. To partially address both these issues, we re-

run the multivariate cross-sectional regressions above with the three annual

tail riskmeasures and the four standard control variables (BETA, SIZE, BM,

and MOM), varying the fraction of the lower half of the return distribution

over which we calculate tail risk. The results are reported in Table 4, which

contains the average slope coefficients and associated t-statistics for the tail

risk measures but omits the coefficients for the other control variables in the

interest of brevity.For H-TCR, the pattern in coefficients as the definition of the tail of the

distribution changes is consistent with the theoretical intuition. The coeffi-

cient and statistical significance peakwhenH-TCR is calculated using the 5%

tail. When H-TCR is measured over the full lower half of the distribution,

that is, when it becomes a downside risk rather than tail risk measure, the

magnitude of the coefficient is small, and it is statistically insignificant.

13 The choice of a longer period overwhich to calculate tail risk allows us to godeeper into the tail, that is, to the 5%level, while still having a sufficient number of daily observations.


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The pattern for stock-specific risk, as measured by LPM(Ri), is markedly

different. The coefficient remains large in magnitude, and its statistical sig-

nificance is preserved for all tail values from 10% to 50%, with the statistical

significance, if anything, increasing as the size of the tail increases.

Interestingly, the coefficient is smaller and not statistically significant for

the 5% tail. This evidence confirms the conclusion above that the significance

of LPM(Ri) is not due to its ability to pick up tail risk at all. Instead, it is

proxying for the more general features of the return distribution, such as

idiosyncratic volatility (IVOL) and extreme returns (MAX). One should

not conclude from these results that stock-specific tail risk is unpriced but

rather that disentangling the pricing of this risk from the pricing of related

distributional risks is an extremely challenging empirical exercise, which is

beyond the scope of this paper.The last row of Table 4 provides evidence for the negative, but statistically

insignificant, coefficients for systematic tail risk (�LPM) across the various

definitions of the tail, except for the 50% tail, where the coefficient is small,

positive, and statistically insignificant. However, the �LPMmeasure estimated

from the lower half of the return distribution (50% tail) is interpreted differ-

ently from the �LPM measures estimated from the extreme left-tail of the

return distribution (5% and 10%). Specifically, the �LPM obtained from the

50% tail corresponds to the downside risk measure of Ang, Chen, and Xing

(2006), whereas the �LPM measures obtained from the 5% and 10% tails can

be viewed as a proxy for market crash risk as they quantify exposures of

individual stocks to the aggregate stock market during large falls of the

market.The insignificant coefficients on �LPM from the 5% and 10% tails indicate

that this proxy for crash risk is not priced in the cross-section of individual

stocks. These results may appear to contradict to the findings of Kelly and

Table 4

Tail risk measures for differing tail sizes

Tail

5% 10% 20% 30% 40% 50%

H-TCR 1.2366 0.9779 0.6194 0.3987 0.4938 0.2596(3.43) (3.15) (2.34) (1.80) (2.75) (1.56)

LPM(Ri) �0.0278 �0.0753 �0.0820 �0.0782 �0.0569 �0.0627(�0.85) (�2.81) (�4.43) (�5.39) (�5.36) (�5.72)

�LPM �0.0737 �0.0826 �0.1046 �0.0697 �0.0878 0.1386(�1.96) (�1.25) (�0.84) (�0.38) (�0.35) (0.44)

We report the average slope coefficients, over the sample period July 1963 to December 2012, and associatedNewey-West t-statistics (in parentheses) on our three tail risk measures—H-TCR, LPM(Ri), and �LPM—fromFama-MacBeth regressions of one-month-ahead returns on tail risk after controlling formarket beta, size, book-to-market, andmomentum, that is, the standard control variables. For the sake of brevity, results for the controlvariables are not reported. Tail risk measures for the 5%, 10%, 20%, 30%, 40%, and 50% tails are calculatedusing daily returns over the past one year.


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Jiang (2013) and Ruenzi and Weigert (2013). There are a couple of potentialexplanations. First, these results may cast some doubt on the robustness oftheir results. For example, Ruenzi and Weigert (2013) investigate primarilythe contemporaneous relation between crash sensitivity and stock returns,not the predictive relation that we examine.WhenRuenzi andWeigert (2013)do turn to predicting returns in the cross-section, their results are muchweaker, both statistically and economically. Alternatively, it could be that,for a variety of good reasons, it is difficult to detect this relation in the data,and therefore that the specific methodology used matters. Kelly and Jiang(2013) andRuenzi andWeigert (2013) use different definitions and estimationmethodologies for crash risk (or crash sensitivity), and they focus on returnseven deeper in the tail of the distribution. Thus, their results can be viewed ascomplementary to those in our study. We provide strong evidence for thepricing of hybrid tail risk, andKelly and Jiang (2013) andRuenzi andWeigert(2013) provide complementary evidence for the pricing of systematic tail risk.

Similarly, the negative and/or insignificant coefficients on downside beta inTables 3 and 4 appear to contradict the findings of Ang, Chen, and Xing(2006). Interestingly, Ruenzi and Weigert (2013) find a similar result whenthey attempt to control for downside risk. We return to this issue in Section6.4.

The clear conclusion from our empirical analysis is that cross-sectionalregressions provide strong evidence for an economically and statistically sig-nificant positive relation between hybrid tail covariance risk and future re-turns, consistent with models that suggest that risk in the left tail of portfolioreturns is priced and that prices are influenced by investors with concentratedholdings in individual securities and positions in more diversified portfolios.The evidence for both stock-specific and systematic tail risk measures is moremixed. In the former case, the results strongly suggest an inability to distin-guish the desired effect from other effects associated with the stock-specificdistribution of returns. In the latter case, the effect may be too small to detectwith any degree of precision using our proxy given measurement issues.

5. Further Evidence on Hybrid Tail Covariance Risk

Given the evidence above of statistically and economically significant pricingof hybrid tail covariance risk in the cross-section of stocks, we now proceed toexamine this phenomenon in more detail, examining first the time-series per-sistence ofH-TCRand then turning to alternativemeasures of tail covariancerisk and tail beta.

5.1 Persistence

As is appropriate for a study of cross-sectional expected returns, wecalculate tail risk over a specific time window (six months in much of the


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previous analyses) and examine returns over the subsequent month.However, in a rational setting, investors should only care about historicalhybrid tail covariance risk to the extent that it predicts future risk.Alternatively, if one views the phenomenon we document as mispricing,then the portfolio turnover associated with a strategy that exploits this mis-pricing is of interest.To examine this issue, we compute the transition matrix for decile port-

folios formed by sorting on H-TCR. Table 5, panel A, reports the average ofthemonth-to-month transitionmatrices for the stocks in these portfolios, thatis, the average probability (in percent) that a stock in decile i (as given by therows of thematrix) in onemonthwill be in decile j (as given by the columns ofthe matrix) in the subsequent month. Because we calculate H-TCR over sixmonths, there is a 5-month overlap between H-TCR calculated in two adja-cent months, which generates persistence in portfolio membership by con-struction. Thus, the diagonals of the transitionmatrix reflect both the overlapand the persistence in tail risk. For the extreme deciles, portfolios 1 (LowH-TCR) and 10 (High H-TCR), about 70% of the stocks remain in theextreme decile in the following month.To eliminate the persistence caused by the overlap, panel B reports the

average of the 6-month lag transition matrices for the stocks in these port-folios, that is, the average probability (in percent) that a stock in decile i(as given by the rows of the matrix) in one month will be in decile j(as given by the columns of the matrix) 6 months later. Despite thehigh hurdle presented by the six-month lag, this transition matrix alsoshows substantial evidence of persistence. Approximately 20% of stocks inthe extreme portfolios are still in the same portfolios six months later, andapproximately 35%–40% of the stocks are in the top or bottom two deciles.In other words, H-TCR predicts both returns and future tail risk in thecross-section.

5.2 Alternative measures of hybrid tail covariance risk

As presented in Equation (9), we have so far used six months of daily returnsand the 10% lower tail to estimate hybrid tail covariance risk (H-TCR) andpredict returns in the subsequent month. To get additional insight into therelation between tail risk and returns, we generate alternative measures of tailcovariance risk and test their predictive power for the cross-section of ex-pected returns.Specifically, alternative measures of H-TCR are computed based on the

daily returns from the (1) 5% lower tail of the daily return distribution overthe past 12months, (2) 10% lower tail of the daily return distribution over thepast 12 months, (3) 10% lower tail of the daily return distribution over thepast 6months, (4) 20% lower tail of the daily return distribution over the past6months, and (5) 20% lower tail of the daily return distribution over the past


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Table

5

Tim

e-series

averageoftheH-TCR

transitionmatrix

Low

H-TCR

23

45

67

89

HighH-TCR

Panel

A:One-month

lag

Low

H-TCR

72.10%

16.33%

4.41%

2.19%

1.34%

0.96%

0.67%

0.51%

0.41%

0.37%

216.20%

47.68%

19.46%

7.31%

3.60%

2.12%

1.43%

0.97%

0.72%

0.52%

34.62%

19.06%

37.64%

19.86%

8.35%

4.42%

2.55%

1.62%

1.10%

0.76%

42.31%

7.28%

19.48%

32.91%

19.28%

8.86%

4.70%

2.65%

1.56%

1.02%

51.45%

3.67%

8.44%

18.76%

31.21%

18.94%

9.07%

4.67%

2.56%

1.32%

61.06%

2.24%

4.39%

8.85%

18.55%

31.19%

18.83%

8.68%

4.31%

2.05%

70.78%

1.42%

2.64%

4.69%

8.99%

18.44%

32.81%

19.16%

8.02%

3.20%

80.57%

1.00%

1.67%

2.73%

4.72%

8.71%

18.70%

36.65%

19.40%

6.01%

90.50%

0.75%

1.08%

1.69%

2.62%

4.31%

8.03%

18.83%

44.37%

17.94%

HighH-TCR

0.41%

0.57%

0.79%

0.99%

1.33%

2.03%

3.21%

6.26%

17.56%

66.81%

Panel

B:Six-m

onth

lag

Low

H-TCR

23.05%

15.83%

12.30%

10.07%

8.44%

7.32%

6.24%

5.50%

5.31%

5.93%

215.34%

14.11%

12.87%

11.39%

10.05%

8.85%

7.74%

6.98%

6.33%

6.32%

312.07%

12.41%

12.06%

11.57%

10.86%

9.95%

9.09%

8.28%

7.08%

6.63%

410.02%

11.14%

11.53%

11.52%

11.13%

10.68%

9.93%

9.02%

7.94%

7.08%

58.18%

9.92%

10.69%

10.92%

11.24%

11.17%

10.78%

9.96%

9.10%

8.05%

67.28%

8.98%

9.75%

10.59%

11.00%

11.17%

11.18%

10.92%

10.23%

8.89%

76.43%

7.85%

8.95%

9.88%

10.63%

11.36%

11.44%

11.74%

11.52%

10.21%

85.81%

6.95%

8.01%

8.89%

9.95%

10.81%

11.87%

12.41%

12.98%

12.32%

95.52%

6.46%

7.18%

8.02%

8.83%

9.82%

11.47%

12.91%

14.60%

15.18%

HighH-TCR

6.29%

6.36%

6.65%

7.15%

7.86%

8.86%

10.26%

12.30%

14.91%

19.38%

Decileportfoliosareform

edeverymonth

from

July1963

toDecem

ber2012


onthehyb

ridtailcovarian

cerisk

(H-TCR)overthepastsixmonths.PanelAreportsthe

averageofthemonth-to-m

onth

tran

sitionmatricesforthestocksintheseportfolio

s,thatis,theaverageprobability(inpercent)thatastock

indecile

i(as

givenbytherowsofthematrix)inone

month

willbeindecile

j(as

givenbythecolumnsofthematrix)

inthesubsequentmonth.W

erollthesamplemonth

bymonth

(hence

thereisan

overlap

inthecalculationofH-TCR),andthe

tran

sitionmatricesarecomputedforeach

month

inoursample,generatingatotalof594matrices.Pan

elBreportstheaverageofthe6-month

lagtransitionmatricesforthestocksin

these

portfolio

s,thatis,theaverageprobability(inpercent)thatastock

indecile

i(as

givenbytherowsofthematrix)inonemonth

willbeindecile

j(as

givenbythecolumnsofthematrix)6months

later.


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3 months. As the calculation window is reduced, we increase the tail size tokeep the number of daily observations used in computingH-TCR sufficientlylarge.Table 6, panel A, shows that, for all measures of H-TCR, the average

equal-weighted raw return increases when moving from the low H-TCR tothe highH-TCRportfolio. The average raw return differences are in the rangeof 0.50% to 0.74%permonthwithNewey-West t-statistics ranging from 2.98to 5.40. The results, both in terms of the magnitude of the return differenceand its statistical significance, seem to improve as the window length isreduced; that is, we usemore recent information, and as we focus on a smallerfraction of the more extreme tail observations.In addition to the average raw returns, panel A also presents the mag-

nitude and statistical significance of the difference in intercepts from theregression of the equal-weighted H-TCR portfolio returns on a constant,the excess market return, a size factor (SMB), a book-to-market factor(HML), following Fama and French (1993), and Carhart’s (1997) momen-tum (MOM) factor.14 The four-factor alpha differences between the lowH-TCR and high H-TCR portfolios are in the range of 0.45% to 0.63%per month and are highly significant, with the t-statistics ranging from 2.92to 4.90.15

In untabulated results, we also calculate the value-weighted portfolio rawreturn and four-factor alpha differences. As in Table 1, the raw return differ-ences are smaller for these portfolios. Interestingly, although the four-factoralpha differences also tend to be smaller, this effect is less pronounced. Forexample, these quantities are 0.43 and 0.49 for the 6-month and 12-month10% tails, respectively, with corresponding t-statistics of 2.75 and 3.51 thatindicate strong statistical significance.These results indicate an economically and statistically significant, positive

relation between hybrid tail covariance risk and the cross-section of expectedreturns. An investment strategy that goes long stocks in the highest H-TCRdecile and shorts stocks in the lowestH-TCRdecile produces average raw andrisk-adjusted returns in the range of 6% to 9% on an annualized basis.

5.3 Hybrid tail beta

Wehave so farmeasured hybrid risk in the lower tail of the return distributionusing tail covariance risk. In this section, we construct an alternative measurecalled hybrid tail beta, which is defined as the ratio of hybrid tail covariance

14 Small minus big (SMB), high minus low (HML), and winner minus loser (MOM) are described in and obtainedfrom Kenneth French’s data library: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.

15 The three-factor Fama-French alpha differences are even larger andmore statistically significant than those thatcontrol for momentum; however, we report only the latter for the sake of brevity.


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http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/


Table

6

Alternative

measuresofhybridtailcovariance

risk

Panel

A:Alternative

measuresoftailcovariance

risk

5%

H-TCR

in12months

10%

H-TCR

in12months

10%

H-TCR

in6months

20%

H-TCR

in6months

20%

H-TCR

in3months

Return

H-TCR

Return

H-TCR

Return

H-TCR

Return

H-TCR

Return

H-TCR

Low

H-TCR

0.65

�0.411

0.66

�0.447

0.48

�0.262

0.56

�0.237

0.52

�0.150

20.90

�0.238

0.95

�0.259

0.86

�0.147

0.92

�0.117

0.85

�0.075

31.04

�0.162

1.04

�0.173

0.96

�0.098

1.00

�0.060

1.01

�0.042

41.13

�0.112

1.03

�0.112

1.06

�0.064

1.08

�0.018

1.05

�0.019

51.13

�0.072

1.13

�0.060

1.16

�0.037

1.13

0.018

1.11

0.000

61.17

�0.037

1.21

�0.015

1.14

�0.013

1.16

0.052

1.13

0.018

71.21

�0.004

1.22

0.031

1.15

0.011

1.19

0.088

1.19

0.037

81.19

0.033

1.23

0.082

1.30

0.037

1.23

0.129

1.26

0.058

91.22

0.078

1.18

0.148

1.27

0.070

1.20

0.183

1.27

0.087

HighH-TCR

1.16

0.156

1.15

0.259

1.22

0.130

1.13

0.282

1.22

0.140

Return

diff.

0.52

0.50

0.74

0.58

0.70

t-stat.

(3.44)

(2.98)

(5.40)

(3.46)

(5.13)

FFC4alphadiff.

0.50

0.51

0.63

0.45

0.55

t-stat.

(3.92)

(3.59)

(4.90)

(2.92)

(3.91)

Panel

B:Hybridtailbeta

5%

tailbetain

12months

10%

tailbetain

12months

10%

tailbetain

6months

20%

tailbetain

6months

20%

tailbetain

3months

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

Low

tailbeta

0.68

�1.776

0.58

�1.408

0.55

�1.712

0.61

�1.132

0.53

�1.500

20.85

�1.122

0.96

�0.918

0.87

�1.053

0.94

�0.662

0.91

�0.855

31.10

�0.842

1.05

�0.690

1.00

�0.772

1.02

�0.444

1.01

�0.572

41.11

�0.661

1.06

�0.534

1.09

�0.587

1.13

�0.289

1.02

�0.380

51.08

�0.521

1.12

�0.408

1.14

�0.442

1.12

�0.160

1.14

�0.225

61.13

�0.401

1.18

�0.294

1.13

�0.315

1.18

�0.037

1.18

�0.080

(continued)


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Table

6Continued

Panel

B:Hybridtailbeta

5%

tailbetain

12months

10%

tailbetain

12months

10%

tailbetain

6months

20%

tailbetain

6months

20%

tailbetain

3months

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

Return

Tailbeta

71.18

�0.289

1.22

�0.182

1.18

�0.190

1.17

0.094

1.19

0.069

81.20

�0.168

1.22

�0.054

1.25

�0.049

1.28

0.250

1.30

0.248

91.25

�0.009

1.23

0.120

1.32

0.143

1.22

0.468

1.28

0.507

Hightailbeta

1.20

0.314

1.17

0.459

1.25

0.546

1.12

0.907

1.23

1.060

Return

diff.

0.52

0.59

0.69

0.51

0.70

t-stat.

(3.36)

(3.61)

(4.85)

(3.08)

(5.24)

FFC4alphadiff.

0.48

0.55

0.55

0.35

0.52

t-stat.

(3.66)

(3.71)

(3.97)

(2.21)

(3.81)

Inpan

elA,decileportfolio

sareform

edeverymonth

from

July1963

toDecem

ber

2012


onalternativemeasuresofhybridtailcovariance

risk

(H-TCR).H-TCR

iscomputedbased

onreturnsfrom

the(1)5%

lower

tailofthedailyreturn

distributionoverthepast12

months,(2)10%

lower


distributionoverthepast12

months,(3)

10%

lower


distributionoverthepast6months,(4)20%

lower


distributionoverthepast6months,and(5)20%

lower


distributionoverthepast3months.Portfolio

1(10)

istheportfolio

ofstockswiththelowest(highest)H-TCR.Thetablereportstheaverageequal-weigh

tedmonthlyreturns(inpercentage

term

s)andaverageH-TCRvaluesineach

decile.T

helastfourrowspresentthedifferencesinmonthlyreturnsandthedifferencesinalphas

withrespecttothefour-factorFam

a-French-Carhart

model(FFC4Alpha)betweenportfolios10

and1,withassociated

New

ey-W

est(1987)adjusted

t-statisticsinparentheses.InpanelB,w

erepeatthesameexercise,exceptthatstocksaresorted

onthebasisofhyb

ridtailbeta,as

givenin

Equation(13),rather

than

onTCR.


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risk to the lower partial moment of the market portfolio:

Hybrid Tail Beta ¼H-TCR

LPMðRmÞ¼

XRi<hi

ðRi � hiÞðRm � hmÞ

XRi<hi

ðRm � hmÞ2

: ð13Þ

This tail risk measure is the same as the measure of �i;LPM in Equation (6),except that the moments in the numerator and denominator are conditionedon states in which the return on the individual stock is in the tail of its distri-bution rather than on states in which the market return is in the tail. Relativeto tail covariance risk, tail beta has the advantage of being normalized to amore standard scale.

We generate alternative measures of tail beta computed based on dailyreturns from the (1) 5% lower tail of the daily return distribution over thepast 12 months, (2) 10% lower tail of the daily return distribution overthe past 12 months, (3) 10% lower tail of the daily return distribution overthe past 6 months, (4) 20% lower tail of the daily return distribution over thepast 6months, and (5) 20% lower tail of the daily return distribution over thepast 3 months.

Table 6, panel B, shows that for all measures of hybrid tail beta the averageraw return increases when moving from the low tail beta to the high tail betaportfolio. The average raw return differences are in the range of 0.51% to0.70% per month with Newey-West t-statistics ranging from 3.08 to 5.24,similar to those reported in panelA for hybrid tail covariance risk. TheFama-French-Carhart four-factor alpha differences between the low tail beta andhigh tail beta portfolios are also positive and highly significant.16 These resultsindicate an economically and statistically significant, positive relation be-tween expected returns and alternative measures of tail beta.

As with H-TCR, hybrid tail betas measured over shorter periods anddeeper in the tail appear to have more economically and statistically signifi-cant predictive power. Of course, as the window length decreases, it is neces-sary to increase the fraction of the distribution used to measure tail risk inorder to ensure a sufficient number of observations to avoid severe measure-ment error issues.

6. Robustness Checks

While cross-sectional regressions are arguably the best way to deal with largesets of potential risk factors, in this section we provide two alternative ways ofdealing with the potential interaction of tail covariance risk with firm size,book-to-market, past returns, downside beta, liquidity, coskewness, idiosyn-cratic volatility, MAX, and net share issuance. Specifically, we test whether

16 As before, Fama-French three-factor alpha differences are uniformly larger and more statistically significant.


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the positive relation between H-TCR and the cross-section of expected re-

turns still holds once we control for these additional predictors using bivariate

sorts of portfolios and characteristic-matched benchmark portfolios.We also

provide results from a screening mechanism to rule out the possibility that

extreme stocks of various types are driving our results. Finally, following

Ang, Chen, and Xing (2006), we present evidence from the NYSE sample

after excluding highly volatile stocks.

6.1 Dependent sorts

For the dependent bivariate sorts our general methodology is to first form

deciles on the control variable, then, within each of these deciles, to form

deciles on the basis ofH-TCR.We then average theH-TCRdeciles across the

control variable deciles to form portfolios with similar levels of the control

variable but different levels of H-TCR. For example, we control for size by

first forming decile portfolios based on market capitalization. Then, within

each size decile, we sort stocks into decile portfolios based on tail covariance

risk (H-TCR) so that decile 1 (decile 10) contains stocks with the lowest

(highest) H-TCR. The first column in Table 7 averages returns across the

ten size deciles to produce decile portfolios with dispersion inH-TCRbut that

contain firms of all sizes. This procedure creates a set of H-TCR portfolios

with nearly identical levels of firm size, and thus these H-TCR portfolios

control for differences in size. After controlling for size, the average return

increases from 0.63% per month to 1.31% per month when moving from the

low H-TCR to the high H-TCR portfolios, yielding an average return differ-

ence of 0.68% per month, with a Newey-West t-statistic of 5.90. The 10-1

difference in four-factor alphas is 0.44% per month, and it is also highly

statistically significant.17 Thus, market capitalization does not explain the

return difference between high and low H-TCR stocks.We form similar bivariate decile portfolios based on the dependent sorts of

H-TCR and book-to-market, momentum, downside beta, short-term rever-

sals, liquidity, coskewness, idiosyncratic volatility, MAX, and net share issu-

ance. Table 7 shows that after controlling for these variables, the average

return differences between the lowH-TCR and highH-TCR portfolios are in

the range of 0.31% to 0.68% per month, and the differences in four-factor

alphas vary over the same range. These average raw and risk-adjusted return

differences are both economically and statistically significant.Overall, the results in Table 7 indicate that other well-known cross-

sectional effects cannot explain the high (low) returns to high (low) H-TCR

stocks.

17 As before, for the sake of brevity we do not report the differences in three-factor alphas, but they are uniformlygreater in magnitude than are the corresponding differences in four-factor alphas.


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6.2 Characteristic-matched portfolios

In Table 8, we further examine whether the significantly positive relationbetween tail covariance risk and expected returns is due to same variablesused in Section 6.1 but based on a different methodology. In the columnlabeled ‘‘Size/BM Adjusted,’’ we report the average returns in excess of thesize and book-to-market matched benchmark portfolios similar to the pro-cedure of Daniel and Titman (1997). In the next eight columns, we includeadditional controls for momentum, downside beta, short-term reversals, illi-quidity, coskewness, idiosyncratic volatility, MAX, and net share issuance.For each additional control, we first perform a decile sort based on thecharacteristic and then based on hybrid tail covariance risk (H-TCR).Finally, we average the H-TCR deciles across the characteristic deciles andreport size and book-to-market matched returns within each H-TCR decile.Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) tail covari-ance risk.

Table 7

Bivariate portfolios of stocks sorted by H-TCR after accounting for control variables

SIZE BM MOM �down REV ILLIQ COSKEW IVOL MAX ISSUE

Low H-TCR 0.63 0.73 0.84 0.78 0.67 0.68 0.72 0.83 0.86 0.72(2.59) (2.83) (3.39) (2.88) (2.59) (2.56) (2.73) (3.54) (3.51) (2.77)

2 0.88 0.93 0.96 0.95 0.90 0.93 0.97 0.97 0.90 0.93(3.67) (3.80) (3.96) (3.69) (3.68) (3.88) (3.98) (4.32) (3.78) (3.80)

3 1.00 1.04 1.08 1.05 1.02 1.04 1.08 1.04 1.05 1.03(4.46) (4.35) (4.68) (4.19) (4.36) (4.49) (4.57) (4.62) (4.57) (4.41)

4 1.07 1.09 1.09 1.13 1.11 1.07 1.09 1.08 1.06 1.08(4.75) (4.75) (4.81) (4.65) (4.88) (4.64) (4.70) (4.86) (4.68) (4.71)

5 1.11 1.13 1.09 1.13 1.13 1.09 1.10 1.09 1.10 1.18(4.96) (5.06) (4.89) (4.87) (5.04) (4.93) (4.90) (4.86) (4.85) (5.25)

6 1.10 1.11 1.13 1.15 1.19 1.14 1.10 1.10 1.10 1.15(5.02) (5.01) (4.99) (5.02) (5.46) (5.14) (5.00) (4.86) (4.88) (5.15)

7 1.24 1.17 1.16 1.19 1.18 1.26 1.18 1.19 1.18 1.19(5.53) (5.38) (5.28) (5.52) (5.35) (5.63) (5.40) (5.03) (5.26) (5.43)

8 1.29 1.24 1.25 1.21 1.26 1.27 1.24 1.23 1.24 1.22(5.67) (5.51) (5.48) (5.66) (5.54) (5.49) (5.51) (5.21) (5.33) (5.38)

9 1.30 1.23 1.21 1.16 1.22 1.31 1.24 1.20 1.21 1.22(5.46) (5.41) (5.21) (5.52) (5.34) (5.61) (5.54) (4.93) (5.26) (5.44)

High H-TCR 1.31 1.29 1.15 1.19 1.27 1.31 1.22 1.22 1.24 1.24(4.84) (4.98) (4.38) (5.67) (4.86) (4.83) (4.75) (4.66) (4.84) (4.82)

Return diff. 0.68 0.55 0.31 0.41 0.60 0.63 0.50 0.39 0.38 0.52t-stat. (5.90) (4.40) (2.62) (3.32) (4.79) (5.11) (4.02) (3.43) (3.38) (4.27)FFC4 alpha diff. 0.44 0.43 0.30 0.41 0.49 0.46 0.41 0.26 0.28 0.42t-stat. (3.79) (3.62) (2.68) (4.90) (4.26) (3.68) (3.64) (2.43) (2.65) (3.59)

Double-sorted, equal-weighted decile portfolios are formed every month from July 1963 to December 2012 bysorting stocks based on the hybrid tail covariance risk after controlling for size, book-to-market, momentum,downside beta, short-term reversal, illiquidity, coskewness, idiosyncratic volatility,MAX, andnet share issuance.In each case, we first sort the stocks into deciles using the control variable, thenwithin, each decile, we sort stocksinto decile portfolios based on the hybrid tail covariance risk over the previous year so that decile 1 (10) containsstocks with the lowest (highest) H-TCR. The table presents average returns (in percentage terms) across the tencontrol deciles to produce decile portfolios with dispersion in H-TCR but with similar levels of the controlvariable. The last four rows present the differences inmonthly returns and the differences in alphaswith respect tothe four-factorFama-French-Carhartmodel (FFC4Alpha) betweenportfolios 10 and 1,with associatedNewey-West t-statistics in parentheses.


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As presented in the first column of Table 8, after controlling for size andbook-to-market simultaneously, the average return difference between thelow H-TCR and high H-TCR deciles is about 0.58% per month with aNewey-West t-statistic of 5.52. The last eight columns of Table 8 clearlyshow that the spreads in size and book-to-market adjusted returns betweenH-TCR deciles 10 and 1 remain positive and significant after controlling formomentum, downside beta, short-term reversals, liquidity, coskewness, idio-syncratic volatility, MAX, and net share issuance. The average return differ-ence between the low H-TCR and high H-TCR deciles is in the range of0.33% to 0.58% per month with t-statistics ranging from 3.87 to 6.38.Thus, the predictive power of hybrid tail covariance risk is not due to theaforementioned cross-sectional effects.

6.3 Further screening for liquidity, volatility, momentum, and reversals

So far, we have used bivariate sorts, characteristic-matched portfolios, andcross-sectional regressions to deal with the potential interaction of the tailcovariance risk with numerous other variables. An alternative way of demon-strating the robustness of our results is to exclude small, low-priced, illiquid,highly volatile, and extreme short-term and intermediate-term winner andloser stocks in the formation of H-TCR portfolios. As discussed earlier, weuse a sample that excludes NYSE/AMEX/NASDAQ stocks that are in the

Table 8

Characteristic-matched portfolios of stocks sorted by H-TCR: Additional controls for SIZE/BM and

MOM, �down, REV, ILLIQ, COSKEW, IVOL, MAX, and ISSUE

Additional control variables

SIZE/BMadjusted

MOM �down REV ILLIQ COSKEW IVOL MAX ISSUE

Low H-TCR �0.40 �0.23 �0.31 �0.40 �0.35 �0.34 �0.25 �0.22 �0.352 �0.17 �0.14 �0.13 �0.20 �0.14 �0.14 �0.12 �0.18 �0.173 �0.07 �0.03 �0.07 �0.06 �0.03 �0.01 �0.08 �0.05 �0.054 �0.01 0.00 0.03 0.01 �0.03 0.01 0.00 �0.03 �0.015 0.07 �0.02 0.04 0.05 �0.01 0.01 �0.01 �0.01 0.076 0.01 0.04 0.06 0.06 0.04 0.00 0.00 0.01 0.047 0.06 0.05 0.10 0.06 0.11 0.06 0.08 0.06 0.078 0.17 0.14 0.09 0.14 0.11 0.12 0.12 0.13 0.119 0.15 0.11 0.09 0.15 0.13 0.14 0.10 0.11 0.13High H-TCR 0.18 0.10 0.11 0.18 0.18 0.15 0.17 0.18 0.17

Return diff. 0.58 0.33 0.42 0.58 0.53 0.49 0.42 0.40 0.53t-stat. (5.52) (3.87) (5.27) (6.38) (5.49) (5.69) (4.32) (4.41) (5.87)

In the column labeled “Size/BM adjusted,” we report the average returns of H-TCR portfolios in excess of thesize- and book-to-market-matched benchmark portfolios similar to Daniel and Titman (1997) for the sampleperiod July 1963 to December 2012. In the next six columns, we include additional controls for momentum,downside beta, short-term reversal, illiquidity, coskewness, idiosyncratic volatility,MAX, andnet share issuance.For each additional control, we first perform a decile sort based on the characteristic and then on hybrid tailcovariance risk (H-TCR). Then we average the H-TCR deciles across the characteristic deciles and report sizeand book-to-market matched returns within each H-TCR decile. Portfolio 1 (10) is the portfolio of stocks withthe lowest (highest) tail covariance risk.We report the average returns inmonthly percentage terms. The last tworows report the difference in returns between portfolios 10 and 1 with associated Newey-West t-statistics inparentheses.


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smallest NYSE size decile and stocks trading below $5 per share. Our further

screening process for liquidity, volatility, intermediate-term return continu-

ation, and short-term return reversals is as follows:

1. Liquidity: To screen for liquidity, all NYSE stocks are sorted each

month by the ratio of daily absolute stock return to daily dollar

volume, averaged in a month, to determine the NYSE decile break-

points. Then we exclude all stocks that fall in the smallest NYSE

liquidity decile (or the largest NYSE illiquidity decile).2. Volatility: To screen for volatility, all NYSE stocks are sorted each

month by their idiosyncratic volatility to determine the NYSE decile

breakpoints for the volatility measure. Then we exclude all stocks that

fall in the highest NYSE idiosyncratic volatility decile.3. Winners: To screen for past 6-month winners, all NYSE stocks are

sorted each month by their 6-month cumulative returns from month

t-7 to t-2 to determine the NYSE decile breakpoints. Then we exclude

the winner stocks that fall in the highest NYSE return decile. We

follow the same procedure for past 1-month winners.4. Losers: We screen for past 6-month and 1-month losers using a pro-

cedure analogous to that described above for winners.5. Momentum: To screen formomentum, we eliminate both the 6-month

winners and 6-month losers as described above.6. Short-term reversals: As indicated by Jegadeesh (1990) and Lehmann

(1990), returns in month t–1 are subject to short-term reversals in

month t. Hence, to screen for short-term reversals, we estimate the

tail covariance risk using daily returns frommonth t–12 to month t–2,

skipping returns in month t–1.

After further screening for liquidity, volatility, momentum, and short-term

reversals, the decile portfolios are formed every month from January 1963 to

December 2012 by sorting theNYSE/AMEX/NASDAQ stocks based on the

tail covariance risk calculated using the 10% lower tail of the daily stock

return distribution over the past 6 months. Portfolio 1 (10) is the portfolio

of stocks with the lowest (highest) tail covariance risk.Table 9 reports the average returns in monthly percentage terms on

H-TCR portfolios. The average raw return difference between deciles

10 and 1 is in the range of 0.32% to 0.67% per month with Newey-West

t-statistics ranging from 2.29 to 4.65. The 10-1 differences in four-factor

alphas are also positive and economically and statistically significant. Based

on the average raw and risk-adjusted return differences, we find a positive and

significant relation between hybrid tail covariance risk and expected returns

after screening for small, illiquid, highly volatile, and extreme winner and

loser stocks.


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6.4 The NYSE sample after excluding highly volatile stocks

As discussed in Section 4, the statistically insignificant and often negativecoefficients for systematic tail risk are evident across the various definitionsof the tail. Like many existing papers, we are unable to document that sys-tematic risk, in our case tail risk, is priced in the cross-section. However, theresults for the 50% tail in Table 4 appear to contradict those of Ang, Chen,andXing (2006) (ACXhereafter), who report significant compensation in theformof higher expected returns for those stocks with higher systematic down-side risk. Our sample period and methodology differ somewhat, but the pri-mary reason for the discrepancy appears to be sample selection. ACX restricttheir sample to NYSE stocks and eliminate the 20% of these stocks with thehighest volatility, whereas we use the broader NYSE/AMEX/NASDAQsample and eliminate small and low-priced stocks due to concerns aboutliquidity and the associated microstructure issues.

Table 9

Portfolios of stocks sorted by hybrid tail covariance risk after screening for liquidity, idiosyncratic vola-

tility, winners, losers, momentum, and reversal

Illiquidity Volatility Past6-monthwinners

Past6-monthlosers

Momentum Past1-monthwinners

Past1-monthlosers

Reversal

Low H-TCR 0.65 0.97 0.55 0.91 0.82 0.80 0.56 0.722 0.91 1.10 0.84 1.08 1.02 0.98 0.88 0.943 1.00 1.10 0.95 1.04 1.03 1.05 0.98 1.034 1.11 1.19 1.04 1.17 1.15 1.13 1.06 1.135 1.16 1.19 1.12 1.20 1.12 1.17 1.14 1.126 1.10 1.14 1.05 1.16 1.13 1.13 1.08 1.147 1.19 1.25 1.12 1.22 1.17 1.22 1.15 1.208 1.28 1.33 1.28 1.31 1.27 1.35 1.27 1.319 1.28 1.29 1.16 1.29 1.21 1.31 1.24 1.30High H-TCR 1.24 1.33 1.07 1.30 1.14 1.28 1.22 1.29

Return diff. 0.59 0.37 0.52 0.40 0.32 0.48 0.67 0.56t-stat. (4.00) (2.78) (3.60) (2.72) (2.29) (3.21) (4.65) (4.03)FFC4 alpha diff. 0.46 0.26 0.44 0.33 0.29 0.36 0.57 0.46t-stat. (3.29) (2.00) (3.39) (2.39) (2.27) (2.55) (4.14) (3.46)

We form equal-weighted decile portfolios everymonth from July 1963 toDecember 2012 by sorting stocks basedon hybrid tail covariance risk (H-TCR), calculated using the lower 10% tail of the daily return distribution overthe past sixmonths, after screening for liquidity, idiosyncratic volatility, momentum, and reversals. To screen forliquidity, all NYSE/AMEX/NASDAQ stocks are sorted for eachmonth by the ratio of absolute stock return toits dollar volume to determine the NYSE decile breakpoints for the illiquidity measure. Then we exclude allstocks that belong to the largest NYSE illiquidity decile. To screen for idiosyncratic volatility, all NYSE stocksare sorted for each month by their idiosyncratic volatility to determine the NYSE decile breakpoints for thevolatilitymeasure.Thenwe exclude all stocks that belong to the highestNYSEvolatility decile. To screen for past6-month winners and losers, all NYSE stocks are sorted for each month by their 6-month cumulative returnsfrommonth t-7 to t-2 to determine theNYSEdecile breakpoints formomentum (orwinners and losers). Thenweexclude the winner (loser) stocks that belong to the highest (lowest) momentum decile. We also screen bothmomentumwinners and losers simultaneously. To screen for past 1-month winners and losers, all NYSE stocksare sorted for eachmonth by their past 1-month return to determine theNYSEdecile breakpoints for short-termreversals. Then we exclude the past 1-month winner (loser) stocks that belong to the highest (lowest) reversaldecile. To screen for short-term reversals, we estimate tail covariance risk using daily returns frommonth t-12 tomonth t-2, skipping month t-1. Portfolio 1 (10) is the portfolio of stocks with the lowest (highest) H-TCR. Thelast four rows present the differences in monthly returns and the differences in alphas with respect to the four-factor Fama-French-Carhart model (FFC4 Alpha) between portfolios 10 and 1, with associated Newey-West t-statistics in parentheses. Average raw and risk-adjusted returns are given in percentage terms.


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ACX test the significance of a contemporaneous relation between down-side market beta, �down, and expected returns on NYSE stocks. Their resultsindicate a positive and significant contemporaneous relation between down-side beta and the cross-section of expected returns. ACX also examine thecross-sectional predictive power of �down for a subsample excluding highlyvolatile NYSE stocks. They find no evidence for a significant link between�down and one-month-ahead returns in univariate portfolios.

In this section, we first replicate the main findings of ACX based on theequal-weighted portfolios of downside beta for their sample period of July1963–December 2001. Following ACX, quintile portfolios are formed everymonth from July 1963 to December 2001 by sorting NYSE stocks based ontheir total volatility computed using daily returns over the past one year. ThenNYSE stocks in the highest volatility quintile are removed. After excludingthe most volatile 20% of the NYSE stocks, decile portfolios are formed everymonth from July 1963 to December 2001 by sorting stocks based on �down.

The results are presented in the right panel of Table 10. The average returndifference between decile 10 (High �down) and decile 1 (Low �down) is about0.22% per month with the Newey-West t-statistic of 1.05. This economicallyand statistically insignificant average return difference is very similar to thefinding of ACX from their quintile portfolios. As presented in the first columnof their Table 8 (ACX, p. 1226) the average return difference between quintiles5 and 1 is 0.11% per month with an OLS t-statistic of 0.60. In addition to theaverage raw returns, we also present the magnitude and statistical significanceof the four-factor alpha differences. As shown in the last row of Table 10(right panel), the 10-1 difference in the FFC4 alphas is about 0.21% permonth with a Newey-West t-statistic of 1.24. These results indicate thatthere is no significant predictive relation between downside beta and expectedreturns even after excluding the most volatile 20% of the NYSE stocks.

Next, we investigate the predictive power of hybrid tail covariance risk forthe same stock sample. After excluding the most volatile 20% of the NYSEsample, decile portfolios are formed every month from July 1963 toDecember 2001 by sorting stocks based on H-TCR. The left panel ofTable 10 reports the equal-weighted average returns and the average H-TCR of individuals stocks for each portfolio, where 6-month 10% H-TCRis estimated based on the 10% lower tail of the daily return distribution overthe past 6months, and 12-month 10%H-TCR is estimated based on the 10%lower tail of the daily return distribution over the past 12 months.

For the 6-month 10%H-TCR, the average return difference between highH-TCR and low H-TCR portfolios is about 0.42% per month with a t-stat-istic of 3.68, and the corresponding FFC4 alpha is about 0.35% per monthwith a t-statistic of 3.11. Similar results are obtained from the 12-month 10%H-TCR measure; the average return and FFC4 alpha differences betweenhigh H-TCR and low H-TCR portfolios are positive and statistically signifi-cant. Table 10 clearly shows a positive and significant predictive relation


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between hybrid tail covariance risk and expected returns for the same sampleof NYSE stocks used by ACX.

7. Conclusion

Motivated by underdiversification on the part of individual investors andfurther evidence that these investors hold some of their wealth in well-diver-sified funds in addition to their concentrated holdings of equities, we con-struct a measure of tail risk that depends on the covariance betweenindividual stock returns and the market return conditional on the returnson the stock being in the lower tail of its distribution. This measure, whichwe denote hybrid tail covariance risk (H-TCR), shows significant and robustability to predict future returns. Annualized risk-adjusted return differencesbetween high and low H-TCR stocks are 6%–9%, depending on the precisespecification and methodology.These results contrast starklywith those from either purely stock-specific or

purely systematic measures of tail risk. In the former case, this risk appears tobe negatively priced, an anomalous result that we attribute to severe colin-earity between stock-specific tail risk measures and other distributional

Table 10

Tail covariance risk and downside beta for Ang, Chen, and Xing (2006) sample

6-month 10%H-TCR

12-month 10%H-TCR

Downside beta

Return H-TCR Return H-TCR Return �down

Low H-TCR 1.04 �0.171 1.10 �0.285 Low �down 1.10 0.2162 1.20 �0.101 1.15 �0.176 2 1.17 0.3853 1.18 �0.066 1.15 �0.118 3 1.20 0.5074 1.23 �0.043 1.26 �0.076 4 1.26 0.6185 1.22 �0.023 1.27 �0.041 5 1.36 0.7236 1.28 �0.004 1.29 �0.006 6 1.31 0.8287 1.35 0.015 1.32 0.030 7 1.35 0.9398 1.37 0.036 1.39 0.069 8 1.37 1.0699 1.37 0.062 1.35 0.119 9 1.27 1.240High H-TCR 1.46 0.107 1.44 0.199 High �down 1.32 1.539

Return diff. 0.42 0.34 Return diff. 0.22t-stat. (3.68) (2.56) t-stat. (1.05)FFC4 alpha diff. 0.35 0.23 FFC4 alpha diff. 0.21t-stat. (3.11) (2.00) t-stat. (1.24)

FollowingAng, Chen, andXing (2006), quintile portfolios are formed everymonth from July 1963 toDecember2001 by sortingNYSE stocks based on their total volatility computed using daily returns over the past one year.Then NYSE stocks in the highest volatility quintile are removed. After excluding the most volatile 20% of theNYSE stocks, decile portfolios are formed every month from July 1963 to December 2001 by sorting stocksbased on hybrid tail covariance risk (H-TCR) and downside beta measure of Ang et al. (2006). The left panelreports the equal-weighted averagemonthly returns (in percentage terms) and the averageH-TCRof individualsstocks for each portfolio, where 6-month 10% H-TCR is estimated based on the 10% lower tail of the dailyreturn distribution over the past 6 months, and 12-month 10%H-TCR is estimated based on the 10% lower tailof the daily return distribution over the past 12 months. The right panel presents the equal-weighted averagemonthly returns and the averagedownside beta of individuals stocks for eachportfolio.The last two rowspresentthe differences in average monthly returns and the differences in four-factor alphas (FFC4 alpha) betweenportfolios 10 and 1 and the associated Newey-West adjusted t-statistics in parentheses.


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characteristics of stock returns such as volatility and the prevalence of ex-treme returns, both of which have been shown to be strongly correlated withfuture returns. This issue clearly deserves further study. In the latter case, theestimated price of risk is often negative, small in magnitude, and statisticallyinsignificant, adding to the wealth of existing failures to detect the pricing ofsystematic risk in the cross-section of stocks.

Appendix: Variable Definitions

Market beta: To estimate the market beta of an individual stock, we assume a single-factor

return-generating process:

Ri;d � rf ;d ¼ �i þ �i � Rm;d � rf ;d

� �þ "i;d ; ð14Þ

whereRi;d is the return on stock i on day d,Rm;d is themarket return on day d, and rf ;d is the risk-

free rate on day d. We estimate Equation (14) for each stock using daily excess returns over the

past one year. The estimated slope coefficient �̂ i ¼ cov Ri;d � rf ;d ;Rm;d � rf ;d

� �=var

Rm;d � rf ;d

� �is the market beta of stock i.

Downside beta: Following Ang, Chen, and Xing (2006), we estimate downside beta of an indi-

vidual stock using daily stock and market returns over the past one year, conditioning on move-

ments of the market excess return below its average value:

�down ¼cov Ri;d � rf ;d ;Rm;d � rf ;d jRm;d � rf ;d < �m

� �var Rm;d � rf ;d jRm;d � rf ;d < �m

� � ; ð15Þ

where �m is the average daily excess market return over the past one year.

Size:Following the existing literature, firm size ismeasured as the natural logarithm of themarket

value of equity (a stock’s price times shares outstanding) for each stock.

Book-to-market: Following Fama and French (1992), we compute a firm’s book-to-

market ratio using its market equity at the end of December of year t–1 and the book value of

common equity plus balance-sheet deferred taxes for the firm’s latest fiscal year ending in calendar

year t–1.18

Momentum: Following Jegadeesh and Titman (1993), the momentum variable for each stock in

month t is defined as the 6-month cumulative return from month t–7 to month t–2, that is,

skipping the most recent month. We also use an alternative definition of momentum over the

previous 12 months, again skipping the most recent month, that is, the 11-month cumulative

return from month t–12 to month t–2.

Short-term reversals: Following Jegadeesh (1990) and Lehmann (1990), the reversal variable for

each stock in month t is defined as the return on the stock over the previous month, that is, the

return in month t–1.

Illiquidity: Following Amihud (2002), we measure illiquidity of stock i in month t, denoted

ILLIQ, as the ratio of daily absolute stock return to daily dollar trading volume, averaged in

month t:

ILLIQi;t ¼1

n

Xn

d¼1

jRi;d j

Volumei;d; ð16Þ

18 To avoid giving extreme observations heavy weight in our analysis, following Fama and French (1992), thesmallest and largest 0.5% of the observations on book-to-market ratio are set equal to the next largest andsmallest values of the ratio (the 0.005 and 0.995 fractiles).


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where Ri,d and Volumei,d are the daily return and daily dollar trading volume for stock i on

day d, and n is the number of daily observations in month t.

Coskewness: Following Harvey and Siddique (2000), coskewness is defined as

Coskewi ¼E Ri � �ið Þ Rm � �mð Þ

2� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivar Rið Þ

pvar Rmð Þ

; ð17Þ

where Ri is the daily return on stock i over the past one year, Rm is daily market return over the

past one year, and �i and �m are the average daily returns on Ri and Rm, respectively. var ðRiÞ

and var ðRmÞ are the variance of Ri and Rm, respectively.

Idiosyncratic volatility: Following Ang et al. (2006), we estimate the monthly idiosyncratic vola-

tility of an individual stock using the three-factor Fama-French (1993) model:

Ri;d � rf ;d ¼ �i þ �iðRm;d � rf ;d Þ þ giSMBd þ diHMLd þ "i;d ; ð18Þ

where SMBd and HMLd are daily returns on the size and book-to-market factors of Fama and

French (1993), and "i;d is the idiosyncratic return on day d. Following Ang et al. (2006), the

idiosyncratic volatility of stock i in month t is defined as the standard deviation of daily residuals

in month t: IVOLi;t ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivar ð"i;d Þ

p.

Maximum daily return: Following Bali, Cakici, and Whitelaw (2011), MAX is defined as the

maximum daily return within a month:

MAXi;t ¼ maxðRi;d Þ d ¼ 1; :::;Dt; ð19Þ

where Ri;d is the return on stock i on day d and Dt is the number of trading days in month t.

Net share issuance: Net share issues (ISSUE) are defined as the change in shares outstanding for

the one year ending in December of the previous year. Following Daniel and Titman (2006) and

Pontiff and Woodgate (2008), we measure net share issues via CRSP, using the change over 12

months in split-adjusted shares outstanding, which is negative when firms on balance repurchase

during the 12-month period and positive when on balance they issue. This approach, also used by

Fama and French (2008), allows us to cover all issues and repurchases, including those not

publicized.

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